Let me recommend Polaris, which is a very odd game but potentially just what you're looking for. It's also potentially not at all what you're looking for, because Polaris is weird, but nothing ventured nothing gained eh?
By default in Polaris, you are knights under the command of a king who has begun to go mad. As a last resort, you have formed a kind of secret coalition among some of your fellows to try and secure the civilization you all love. You are also doomed to fail. I suspect changing the fluff so that you're spies in the house of a foreign king will not change the core system too much. (Though you don't have to- there are other remnants, and probably dukes or the like you could spy on. Also, I suspect that the Knight's relationship with their king is adversarial enough to work for what you want.) If you do not want to end in tragedy, that's a bit harder.
Has a mechanic to handle intrigue and influence of PCs
I'm not entirely sure of your meaning. If you want something like D&D 3.5's Attitude, Polaris doesn't exactly have this. If something more like Fate's aspects would do, Polaris almost gets there. Protagonists have themes and aspects and even cosmos that can relate to people, but you never really attach numbers to the question of how much an NPC likes you.
However, if you're comfortable with not having numbers attached to relationships, Polaris's mechanics transition smoothly from duels with blades to duels with words. I particularly like the negotiation aspects, as there's a sort of four way tug of war between the protagonist, the NPC, and the guides. It works really well, though it takes a few readings to be comfortable with it.
On the other hand, given the way that each protagonist has different guides (I'm controlling Urist McProtagonist as his Heart, but Urist's Mistaken is a different player and controls all the antagonists in his scenes, and Urist's Full Moon is a third player and they control about half the NPCs while the other half is controlled by the fourth player, called the New Moon) interplayer influence is fascinating, and by breaking up the DMing role like this you often wind up with very NPC focused challenges.
Simple and light on dice rolling
Polaris is not diceless, but it comes very close.
Whether it counts as simple is up for debate. The entire rulebook is about sixty pages, half of which is fluff. The core system involves the use of key phrases, like "But only if" or "And so it was" that have specific meanings in the context of the game. For example, "And so it was" begins and ends scenes, "But only if" is a negotiation with another player, allowing what they said to be true if they accept the addition you're making. Polaris is honestly fairly simple, but tends to confuse experienced RPG grognards because it doesn't really use any of the standard mechanics of these games. (There's no real hit point equivalent, there are no dice, there is no DM...)
Handle combat and mystery
Combat is easy. Polaris can do swordfights, though as you may have gathered they won't play like anything you'd see in D&D. (Even though they might look a lot alike in-universe.)
By mystery, do you mean that the player is unaware of something and trying to find out, or the character is unaware of something and trying to find out? The latter is easily accomplished, and will probably run smoother than most mysteries you've played through. (The wonderful scene setting rules are perhaps the least noticed genius of this game.) The former could theoretically be done, but would probably require some fast and clever play on the part of one of the guides.
Low Fantasy, Dark Fantasy or Gaslamp Fantasy settings
The main setting for Polaris is dark fantasy. The game is subtitled "Chivalric Tragedy at the Utmost North" and invariably ends badly for all concerned. The end of the world is coming, the end of an age of beauty and light and majesty, and demons and dawn both rise to end the world as we know it.
Mechanically, I can confirm the system works perfectly as low fantasy, though be very sure that all players are on the same page as to what is appropriate. If deviations occur during play, roll with it- whoever's Guide it falls under at the moment has say. My playgroup ran a Lies of Locke Lemora-inspired Polaris game, and it worked wonderfully. (Bug lived! Sweet, sweet victory!)
No idea on Gaslamp Fantasy. My instinct is that it might work, but gaslamp might also trend too optimistic for the tragedies that Polaris produces.
Cloak and dagger, several-layer conspiracies
Read the section on Mysteries again for the distinction between player and character, and how Polaris handles that.
The mechanics do not prevent cloak and dagger; There's no equivalent to a sneak attack, but it will usually help in negotiations. Run as close to default as you can, the game starts out with on conspiracy (the Knights against the King) and will almost certainly have a conspiracy within a conspiracy before it ends (as one or more of the Knights turn against each other.) Add to that demons of the heart and soul, which are invisible, can possess people, easily penetrate your fortifications, and work for the downfall of all you hold dear (did I mention some can destroy your memories of love just for shits and giggles?) and you have a recipe for some pretty solid paranoia.
Reward players for successful scheming
By clever play and skilled negotiation, the things you want to happen will be brought about. By collusion, appeasement, and outright bribery of your Mistaken, you can greatly extend the lifespan of your character. Does that count? No?
How about this; experience is checked when you act in sympathy with the demons, in hatred of a person, or in despair or apathy towards the people as a whole. In other words, by fundamentally betraying your team, you get XP. (Of course, if you get too much XP then you completely betray your team and stop being called a protagonist. You can still play the character in scenes though.)
TL;DR
I don't know if this is the best fit for what you want. It's a weird game, and requires a kind of solemnity of play that isn't what everyone's looking for. But it's a great game, and if you want a courtly feel I can't recommend anything better. It fits most of your criteria, though often in unusual ways. Give it a look.
