Erechel
Explorer
It is very possible to play a political campaign, and you can even use the existing rules without too much tweak if you focus on political encounters, though they are pretty much scattered throughout the books.Before I leave more specific advice, 5e might not be the best system for this - would you consider running something else or are you locked into 5e?
As for "caster supremacy" here, it is nonsense. Charms are pretty obvious: anyone targeted by a charm will know after a minute or so, which isn't at all the common span of any social interaction. Besides, most spells are obvious when casted, and so trying to charm a king will become an unbalanced combat pretty fast. More so, any decision under a spell influence will be ignored. Even minor buffs as Guidance will be banned.
A bard, though,will be pretty OP, as it has both the proper main stat and Expertise without having to cast anything. I suggest that most encounters don't rely in one check, but several, which vary in difficulty depending on the approximation (intimidation will be easier with a peasant than with a King). A few useful rules in the DMG and PHB:
a) Renown: You could easily use it as a prerequisite to interact with a important figure, or at least to be taken seriously. You could gain reputation by winning tournaments or arguments. You could also use it as "social HP". NPCs could also have reputation, which could be destroyed or increase difficulty in checks when you interact with them. (DMG pp. 22-23, 78, 129)
b) Resolving social interactions: NPC Attitudes and difficulty of basic Charisma checks are detailed on the DMG, pp. 244-245. You could use Renown (as above) to change attitude or modify the check difficulty. A basic increase would be in one step (5) for each rank in a faction. So, an interaction with a world figure (like a Zhentarim dread lord) will be fairly difficult, even for very small concessions if they are friendly (20). Treat each rank in a faction as if they were nobility ranks: 1 for free men, 2 for low nobility and knights, 3 for intermediate nobles, 4 for high nobility (think dukes and princes) and the fifth rank for Kings and other world-class leaders.
c) Influence (DMG p 78) could be used to auto succeeding in a check or interaction,or as advantage in a check. Track it as inspiration with a person. Even more, you could use it as favors you own. Pulling a favor is a risky endeavor. You may ask a little one or a big one, but a big favor will usually put you in debt or be interpreted as extortion.
d) Lifestyle (PHB): a beggar in a Duke's palace will be thrown away, but a wealthy character will be welcomed. Nice clothes, jewelry and other attributes of the wealthy will attract sycophants, and an Aristocrat lifestyle will bring your character to the forefront of the intrigue.
e) Alignment: knowing the alignment will be invaluable in any interaction, as it allows you to predict certain behaviors, and think who you can trust. This could be inquired via ideals.
f) Backgrounds: Another invaluable resource in a political campaign. Military types will be more prone to help a soldier or a knight, whereas a noble character will have many doors open. Even criminal or urchin backgrounds allow you to contact the streets, etc.
All in all, there is a lot of ground you could cover with these rules. My advice is that you integrate the rules with the campaign, and you include sessions with many elements of intrigue, but don't forget exploration and a small amount of combat to have all players involved. Track things also, so as the players have the sensation of expending resources. Money, position and influence will have a lot more importance here than in your typical dungeon crawl.