Two important considerations:
1. Argument HP shouldn't go up every level.
Physical HP go up every level because everything starts doing higher damage as the levels go up - monsters, magic items, magic spells, etc. This is pretty much the intense design work of most D&D games - figuring out how to scale things as they go up without messing up the math.
Not only that, but there's also the odds that you might end up with something like arguments that drag out forever, and the arguments become extra boring when people run out of new ideas to say.
Social combat limitations/setting stakes
There needs to be a clear rule about what the stakes are for any social conflict. There's some things that just will never work - "I convince the king to give me all of his money!" etc. Vice versa, the players have to know what the consequences are for losing and to be willing to go with them before they pick up the dice.
The Burning Wheel RPG generally does a good job of dealing with this - there has to be a quick agreement about stakes before the social contest begins. If the goal of the opposition is too outrageous, there's no social contest, but once you've agreed to it, then you accept your character might be convinced of it.
So with both of those in mind, maybe something like:
Argument HP is 12 + Wis Bonus
Every round, you make an argument/point/say and do something to win someone over - roll a D6 + (Cha bonus for persuasion/make an impression, Int bonus if intellectual argument). If you have a reputation or greater authority, you roll a D8. If you have massive authority difference (Duke vs. citizen) you roll a D10.
Once you hit 0, you get one last argument/point in return. If both sides are 0, a large compromise has to be made, favoring the last one to hit 0. If the winner still has argument points, BUT is reduced to 5 or less, they have to make a small compromise.