You need to talk to your players
If this is happening both in and out of character, then this is clearly an out-of-game issue that must be handled out-of-game.
Before going into this, I strongly recommend reading this article on the Five Geek Social Fallacies. You shouldn't be tolerating awful or abusive behavior, just because it's being done in a gaming setting.
So, how do we start?
This is not a Rules issue, it is a people issue
If players are being intentionally antagonistic towards one another, and they aren't all having a good time with this (hey, PvP is fun sometimes), then this is a problem. This is NOT a problem you can solve by enforcing ingame rules, because that won't actually fix anything.
"We don't like Player A. But the DM won't let us Attack Player A directly. So we just won't protect Player A and will try to get their character killed in other ways. Like pulling back when the monsters charge, or not healing them, or chucking a rock at an ogre then Stealthing."
If the Barbarian wishes harm to come to the Bard...they will find a way to do it. If you start forcing rules on them all you will get is bitterness, and a never-ending loop of players circumventing your rules, or just rage-quitting. And the process leading up to it will make everyone miserable.
You need to sit down and have a hard discussion about what is actually going on. D&D is a social game that is intended to allow a group of people to all get together and have a good time. It appears that your Barbarian is not on board with this idea. You need to find out why.
Does the Barbarian's player think his character just doesn't like the Bard's character? (See: My Guy Syndrome) Does the Barbarian's player dislike the Bard's player for some reason? Is the Barbarian's player just a bully?
You need to talk about this like adults. What's going on, how do we resolve it? Because this is not okay as it is. D&D is intended to let us have fun together, and the Barbarian is making the Bard miserable.
If the Barbarian refuses to stop, you have a bad player. You have made it clear that he is making another person miserable with his actions, and he didn't care. So the resolution is simple: "If you can't play nice, you can't play. Goodbye."
Nobody likes to have to expel a player from their game...but it's better to get rid of the one bad egg than for them to ruin the game for everyone else (including you).