So much material for excellent roleplay here !
- First off, really, seriously, talk with your DM outside the game and work it out.
Ultimately, what the DM says goes in his campaign, so you'd better be prepared to convince him. Having your PC's abilities completely crippled because of an obtuse DM jumping at the chance to unfairly punish a player, because "reasons", that just ain't fun at all. And fun is the main goal of the game after all.
Also talk, one by one, to the other players, completely out of the game context. Ideally, meet up at some restaurant or other venue where you can talk at leisure. At worst, do it by phone or video call. Most of all do not use texting or emails, as such kind of debate needs better bandwidth than mere text: you need to see the face, hear the tone of voice, etc. Tell them about what you want in the game and ask them how they think they can help with reaching that goal.
After the discussion with the DM, maybe have your paladin voice some concerns, and seek the counsel of a local (but amenable) priest of your religion. Try to drag the other group members into that in-game meeting. Once it's clear that it is a mandate from superior authority, your group will all be on the same page about whether it's ok or not to just go kill off everything based on reputation alone, all murder-hobo style.
If the group of PCs is already in front of the giant king as of the end of the last game session and you have only until next week to sort this out to avoid a "total big fight right off the bat early next game session", then you obviously need to skip that last step.
Basically, you first convince the DM (out of game), then each player (out of game), then the whole group "consolidates" the final choice (in game).
You can also capitalize on some of these aspects:
PCs are "atypical" members of their race. Otherwise, they would have stayed home, not gone on adventuring! Even when at home, the various members of a race do not all share the exact same monolithic cultural values "block" of attitudes. It's only a racial tendency, not a law-stick used to whack at a specific creature with in order to dictate it's behavior by the DM. So, really, enforcing racial cultural bias to a PC, that is the same as putting a straight-jacket to a PC, and then putting that inside a block of hard cement. You could say in character that "they" are patronizing racists, that "their" ideas is how they think your noble race really is. If they don't correct, turn back at them every cultural stereotype of their own race "and how their character should be somehow obligated follow it". Nope, elf cannot act hurriedly, ever, all good elves take the long view, always, so no min-maxing your daily hours, it's so totally out of character! Yeah, that is stupid of course, but that is just to show what you are doing by trying to justify your desire to murder a legitimate king, based on hearsay alone, with using your entire race as their excuse. No thanks.
Your dwarf's has a very high respect for the value of life. An evil person, if allowed to live, can be led can learn to change its ways. Killing it instead just seem to "fix" that evilness, but ultimately only go to strengthen the armies of hell. Because, where do you think all the demons ultimately come from? Dead evil souls. It is better to redeem someone evil, than kill him. Who'S to say that this kind of killing won't lead to even worse results? Some other giant will become the giant king. A giant that may feel very justified in starting a war of revenge against the races of those who murdered in cold blood their previous king. There are political ramifications after all.
Explain that those people killing off everything because they merely "see" it as evil, are actually those that are doing evil's work themselves. Evil is disrespectful of life, while Good tries to save lives. That king treated the group justly and fairly to date, are you really going to disrespect that? If you were like all the other elves, you'd have stayed in your forests with all your other tree-lovers. But you are not. We don't judge you by your race or appearance or hearsay alone, because that would be unfair to you. You are judged by your honor in the face of your actions. So grant that lawful king the same treatment as you, yourself, would deserve in his place. His cause is just. And he has shown no trace of being dishonorable. Yet. So, we will hold our hands. For now.
Lawful Good is not only about "fighting" evil beings. Heck, goodness itself isn't about fighting Evil, it's about being Good, and Lawful Good is not all about being Good too. Because if that were the case, then the character would be Neutral Good instead. Lawful Good is about Lawfulness too, in equal parts to Goodness. And here we have a mission that will help restore order. Convince the other PCs that if saving that giant's daughter can convince her and her father to change their ways afterwards, they could become a force of lawful order in the area afterwards. While not truly good, that would still be an improvement compared to before. Here, it is enforcing the rule of law, which in this case "trumps" good. A region falling into chaos would only lead to even more evil in the long run, after all.
Play the "we are better than our races and origins" card: "Somebody has to try to stop the racial hatred between dwarves and giants somewhere, and this is a golden opportunity to do so! Who better than us, heroes, to finally change history for the better?" Meanwhile, attacking the giant king (who surely had to step on his own pride a lot before calling for help a group known to have a dwarf in it!), when it has come to us in fairness to ask for our help, would only confirm their own hatred of our races. They would be right, and we would be the ones in the wrong, here. Are we vile traitors, that we would not only turn our back from a call for help in a time of crisis, but be the first ones to draw our swords and turn all violent against them? If yes, then ask yourself, which side is truly doing something evil here?
The Evil King is currently an ally vs the demon forces. Sure, my friends, you expect the giant king to betray us after we save his daughter, right? Well, that would be very stupid from a king! A king doesn't want to spread out the message that he will quickly betray whomever comes to his call for aid. He would never have become king otherwise. Nobody would want to help him rule afterwards, as his word would not be worth a damned thing. There is this skill called Diplomacy, and it is what allows all the kingdoms to live in "relative" peace, without constantly trying to kill each other off, no matter what their cultural differences might be.
Focus on showing that labels such as "good" and "evil" are for little simple minds. Every race generally thinks it is the best one, the good one, and that all other races are bad and evil "enemies". Look at reasons to act, not act because of appearance or reputation alone. Probably, according to the giants,they are good ones, and it is the dwarves that are "evil". Attacking a king during a parley is a good way to prove that the group are actually the bad guys that want to start off a bloody war.
Currently the greater threat that we must face in the quickest way possible are the demon forces. Deciding to fight the king here, that is doing the demon's work for him. Let's free the daughter first. As part of our service to him, we also negociate some kind of long-term deal with the giant king, in front of his people, for the benefits of all races, some long-overdue lawful reforms. We do not need gold, but having the giant race as a whole becoming more "civilized" instead, that is a rare opportunity, and the hope of longer-term peace, that is worth more than a thousand full plates and swords. If the king stupidly betrays us afterwards anyway, then he will have shown that he has no honor at all, and thus that he is not fit to be the giant king, while we will have proven that our honor is solid and true, and that we will not have struck the hand that was offered to us first.
I'd you're playing D&D 5e, make sure to note to everybody that enforcing strict alignments interpretations stopped being a DM's whipping-tool, and was replaced with a focus on fun instead of a huge restraining order on roleplaying. We're not in the 1990's anymore, get with the program and all that. Alignments are almost gone from D&D 5e, and that is for good reasons. Alignments are useful "behavioral hints" for beginners people that don't yet know how to roleplay, but should not be straightjackets for advanced roleplayers. Also, a character is never "always" of a specific alignment. It's more like a big average than absolutes. Forcing "clear alignmkent" all the time, always, makes charactwrs way more one-dimensional and boring, instead of fleshed out. For example, a generally very good, very generous, and very caring person, could still hate with all his soul all the members one specific particular race, killing them off at every opportunity. Is he good? Is he evil? Alignments are just a starting point, not the be-all end-all of a character's personality. both for PCs and NPCs, it is always much better to describe them in terms of values and goals and motivations, than with an alignment. Otherwise, they end up being cardboard cutout characters, good only for a murder-hobo campaign style.
Maybe you have a group of murder-hobo players (and DM) afflicted by a black-&-white mentality (we "good", we "deserve" all the rewards, everything else "bad", needs to be killed off asap by using lots of cool combat abilities). In that case, your character, with his much more "nuanced" personality, just won't fit in that group, or in that campaign, no matter what you do. Ask the players what kind of game they really want to play: black and white murder-hoboness railroaded-by-DM-so-that-every-problem-is-almost-immediately-ultimately-solved-by-battle, or something with more nuanced and varied personalities, problems, and solutions.
And if the DM lets you solve some problems "not by battle", but only to have the same problems crop back up again later on, until they are "finally" solved by battle, then those "not by battle" events do not really count.
I'm not talking about traps or puzzles or the like, I'm talking encounters with creatures. If your DM has you solve over half of your problems by battles, then either he's got a severe lack of imagination, or... he's a murder-hobo at heart. NPCs aren't there just to be foes to kill off. It makes for weak, boring stories when they are only destined to be killed off.
- If your players just know that the DM will, of course, have the giant king betray the party at the worst possible moment, at full force too (with his leveled-up-by-the-demon-daughter plus his entire giant army, too, while the DM's at it!), then yeah, it becomes a matter of self defense and survival. A preemptive strike is the good approach in that case. Surely not Lawful. Surely not Good. More like Murder-Hobo Chaotic-Survival mode. So, if that is the reason, present the problem, out of character, to the group (ideally, talk one-on-one with at least a couple of them first, out of the game, to reach some kind of better mutual understanding and consensus, before presenting the problem to the DM as a group). Ask if that is really the way they want for the campaign style to go. Once the group is convinced, go, in group, directly ask the DM "Are diplomatic solutions, doomed to eventually fail anyway, almost as if that was decided in advance? If that is the case, then what is the point of presenting such encounters in the first place? If most roleplay with NPCs end up leading to betrtayal or battles, then we will start treating every NPC the same way: avoiding most roleplay altogether, killing off every "potentially hostile" NPC before it can do or say anything. Is that really the type of campaign and of roleplay the DM wants, a feeling of "the entire universe is always going to end up against us anyway, roleplay seems to be a useless choice after all, so let's just skip it and roll for Initiative right away instead? We won't be emotionally invested in those NPCs, because we'll just attack as soon as it is reasonable to do so, no matter what your king says. To be emotionally invested in roleplay, we must know that the roleplay approach will work (and in a mostly permanent way) at least a good chunk of the times. Not super-duper-rarely or "only until he can betray us". Otherwise, we might as well go full battle anyway right away, and totally skip all the roleplay, because obviously it seems that it would be a complete waste of time and efforts. So, which campaign style do we really want to play in?
If the verdict is that, yes, roleplay should remain important, then you've got good grounds upon which to solve the current giant king situation diplomatically, instead of "treasonously" suddenly attacking him "because evil".
Some DMs have a very scripted general story arc for the overall storyline. No matter what you do, for example, win or lose, solve stuff by talk or by wits or by favors or by murder, the next big event is going to be the exact same thing. Present your point that a campaign that does not follow that simple cheap trick is a campaign with a much better structure. If the overall impression of the performance of the heroes is good, why the heck should the giant king decide to betray them? They filled their contract and they proved useful to him! They might again be useful in the future. That giant king is lawful, right? Unless that giant king is a murder-hobo himself that would love to see a big war starting between his giant races and the smaller races. But then there should be rumors and hints that the meeting is just an excuse for the giant king to get rid of them and start hostilities with their races. Also, is the giant king really, really stupid? By sending them against the demons, he is testing their strength. If they fail, he gets rid of a potential obstacle to his growing power without soiling his own hands or spending any forces. If they succeed, he learns that they are powerful enough that, had he decided to "betray" them sooner, the heroes would probably have managed to kill him off! Better to have them as allies (unless that king has a very moronic death wish or something like that).
Present also the point that villains that aren't all-full-evil-wholly-detestable-and-needs-to-be-killed-off-immediately-in-all-situations, but that have actual logical reasons to act the way they do and do the "bad" things that they do, are much more interesting than murder-hobo-targets-1D-cardboard-villains. In this situation, the giant king's impression of the performance of the heroes should be a major factor in determining how the whole political situation of the entire region that will slowly unfold for decades over. Especially in relation to the dwarven race. It really is a golden opportunity for inserting politics and tons of roleplay into the campaign, and reducing this to a "we should kill him before he betrays us" is a pity. "As a gaming group, we can do so much better than just "Let's just kill him because he is evil". Killing someone because you believe he is evil, is an evil act. Unless you have an actual reason, it is just not right. That is like killing off every half-orc PC (even if he's actually a good paladin!) "because everybody knows orcs are evil after all". Judge someone by his honor and actions, not by gross racial generalizations.
If the party still insists to betray the giant king's invitation by attacking him "just because he's bad", word should get out that they have zero respect for a white flag of truce. It should be a big stain on their reputation. Telling the giant king "No, we will not help you, because you're a bad guy according to our races. We will leave here in peace, as you invited us so, but next time we meet, it will be as enemies" well, that is one thing. But using the invite to suddenly attack, that is extremely treacherous and dishonorable. And if the rest of the party doesn't care about reputation and honor at all, well, yeah, murder-hobos they are.
- Also, the DM should not even allow the PCs to "win" a battle vs the giant king. he's inviting the PCs in his inner sanctum, where he controls the environment, were his forces are strongest. The evil king should have some kind of idea about what kind of powers they can bring to bear in a battle, how "honorable" they are, and should not expose himself to a potential defeat. He'd have hidden cards (traps, etc.) and make sure that if things turn bad, he'd have the upper hand anyway. And if he doesn't know the party well enough, then he'd "test them" first before agreeing to any meeting, sending bandits or NPCs or whatever against them from afar to test both their mettle in battle, and their honor and personalities.