There was a game once where the a player was frustrated because his sorcerer character wanted to join a friend in a bar fight but the fighter wouldn't let him and grappled him. The DM only let the him break the grapple in the usual way. The grappled player kept 'roll playing' and was pleading with the fighter (in and out of character) to let him join the fight but the fighter wouldn't submit.
So, this scenario can be frustrating on multiple levels, not just in social situations. The fighter character was taking away player agency(by not letting the sorcerer join the brawl) by using combat tactics and the sorcerer was unable to use his social skills to convince the fighter to let him go because social skills used against a fellow player take away player agency and a player should get to decide what his character does and doesn't do. Yet, the sorcerer really should have been able to persuade his comrade to let him go...
There was lots of frustration post-game.
If I were to have DM'd this game I would have let the sorcerer resist the grapple with persuasion rather than use acrobatics/athletics with the justification that he was convincing the fighter to let him go.
So, in the scenario the OP described, if you are doing PvP over differing opinions:
I'd first ask the players if it was important enough to start a conflict over it.
Then we'd hammer out the stakes (what happens if the barbarian loses and what happens if the bard loses)
Then I'd allow them to use whatever skills and stats were appropriate to the narration.
Then I'd make them do an extended test against each other with the agreed upon skills
This way the social contract is maintained in advance, all mechanics (social and physical) are equally important and the players have agreed to the stakes.
So, if the bard uses persuasion and charisma, I might allow the barbarian to use intimidation and strength to resist.
Bard, "You should help me. Trust me, it'll all work out fine!"
Barbarian, "I can maybe help you out, but if I do and it goes south, I'm going to break every bone in your body"
If the barbarian wins, it's not because the bard failed to persuade but because the barbarian intimidated the bard and got him to back down.
Like any fight, I wouldn't make it one single roll (combats are often several rolls). I might make it three using various skills with the person to win two of three being the victor.