Carefully select and use your spells, which by the rules is undoubtedly the best way to go about mitigating a lack of armor without creating an entirely new game mechanic to compensate for a lack of heavy armor types.
Your Smites are probably going to have to become secondary to survivibility, though thankfully some variants of Paladin are very good at using spells to this effect. If you really want a non-Dex, AC-centered fix to a lack of Heavy Armor, the best solution I can think of is to argue for more "realistic" medium armor. With Breastplate banned the most obvious example of what I mean is also removed, but I'll use Scale Mail as an example. Aside from the armor's own leather backing, it would probably be the norm for trained fighters/paladin to wear some sort of padded armor beneath medium armor, if for no other reason than to prevent the armor itself from causing the person wearing it damage, as would happen if an arrow pierced the armor and carried metal fragments from the armor with it into the skin (more common historically with chain links, but hopefully I made my point). Of course, this second layer of armor would also have the benefit of increasing overall protection- A higher AC. Because of the DM's wish to avoid heavy armor, the trade off for this under layer would be that it only improves the AC by 2, maintains the weight of Padded Armor (8), and, if it was me making the request, I'd suggest wearing this second layer make the Max Dex AC bonus 0 by default, with the ability to equip this second layer corresponding to proficiency with Heavy Armor. This effectively eliminates the classes focusing on pumping up Dex from abusing the feature/option, and can turn Scale Mail into a somewhat weakened version of Chain Mail that should be able to hold a non-Dex centered Paladin over for a while.
It won't make up for a lack of true Heavy Armor in the long run, but if you optimize towards Str/Cha and not Dex (and it does sound like you're wanting to do that) it should help in the first few levels while you pick up what magic you'll be relying on. If the DM goes for the idea... well, I can't guarantee that, though it should satisfy both the aesthetic and combat/lethality reasons for removing Heavy Armor. If the DM shoots you down here, my best bit of advice would be to request a suitable replacement feat/skill/spell/etc. (completely down to how you want to play) to make up for the lack of Heavy Armor in the setting. The ability to use Heavy Armor is heavily figured into the balance of classes that get it, so with its removal something should be inserted to attempt to compensate for such a gaping void. As a DM you'd probably agree to a thoughtful, meaningful feat meant to replace a basic class feature that is, for whatever reason, missing. I imagine your friend is no different. By removing Heavy Armor he's already shown he is willing to flex the rules.
*Note- Was re-reading this after stepping away for a bit before posting, and if I just re-processed the party composition right you're a paladin front lining for a rogue and a ranger in a game focused on "frequent" and "lethal" combat. A 3 man in which the front line character is also probably the main source of any healing, the setting does not allow heavy armor, and the other two players are new and playing... nuanced... classes... Oh boy, you are really in for it. It could be a lot of fun, but if I were in your shoes I'd seriously consider being quite cheap and highly optimized in the build... Deleting the rest of what I wrote in favor of a suggestion.
With your party composition and your DM's heavy armor rule I'd be interested to know if you're going to be allocating or rolling your stats. Ideally, if I were in your shoes, I'd like a naked CHA 18 (+any racial) and a 16-18 DEX and CON (+ any racial). Run to either Oathbreaker (if allowed) or probably Devotion Paladin level 6, give up on trying to equalize the lack of armor and see if I could get Polearm Master in exchange for a major class feature missing. Switch to Warlock for a few levels, preferably with Undying Light patron (? I think I recall it right) if allowed, if not, Fiend. Depending on how things are going I'd probably go Tome/Book Warlock, and raid a ton of spells from other classes for utility. Through this Warlock you can get Devil's Sight (useful in your position with the Darkness cheese combination- cheap but it might really save a pair of new players and not put your unarmored hide in the line of fire to do it), Shillelagh (quaterstaff + Polearm Master + Shield), and I can't recall if Green Flame Blade was in the regular Player's Handbook or UA, but that was fairly nice too. Even Fiendish Vigor, if you're not using Armor of Agathys, might be worth grabbing if you went Warlock a little earlier, as every 5 temp HP it can soak is 5 less you have to try to heal. You might also be able to grab Hellish Rebuke, Armor of Agathys would be worth its weight in gold, or, if Undying Light is allowed, you get bonuses to radiant damage, empowering Smites that can be used on short-rest Warlock slots. Eldritch/Agonizing/Repelling Blasts are always an option, too. Get your Warlock ASI and then go back to Paladin. I know you said you don't want to become some hybrid monstrosity, but it is via Warlock that I remember so many little utility and especially pseudo-tanky, punish-on-hit type spells that could let a lightly armored character hang out in the front line for a bit. It is basically a way of plugging a big hole left when Paladin has Heavy Armor taken away that does not really change the melee focus of the character, instead adding the potential for more smites, more utility, and without forcing you to build on another score. In fact, if you cherry pick the right combination you could probably wind up needing CHA and CON and being able to work around DEX, though I haven't read the classes in a bit so I'm a little fuzzy on the real optimization... But if you can get to slots of a moderate level casting Armor of Agathys with them could be the solution to all (most?) of the naked Paladin's personal problems, while an influx of ritual spells, ranged cantrips, and general skulduggery helps keeps the new players alive and going despite the caster being on the front line and deprived of his usual gear. Doubly so if your DM allows Oathbreaker Paladin.