Yes, the barbarian does have to obey.
As best he can however was added by the DM and even then, it means that, to the best of his ability, the barbarian will attempt to do exactly the spoken command as verbatim as possible. For example, if the barbarian needs to climb a wall to reach the bard, then he will do so. He has to do his best to go attack the bard, after all, because that was the command.
What "as best he can" doesn't mean is that the barbarian will not only do his best to successfully try to perform the command, but also do it in the best way possible to do so, going even beyond what the stated command said.
The barbarian will do his best to be able to do exactly what the command said: attack. Yes. But doing the attack the most powerful and best way that he can, not necessarily. Probably "attack the bard" could be interpreted as "simply attacking" that the bard, attacking like the barbarian usually attacks.
Example: if the barbarian is in the habit of doing all the attacks he can, but not raging unless the situation seems serious, then as the barbarian knows he's much stronger than the bard, he'd do his two attacks, but without raging.
If the monster had commanded "Attack the bard to the best of your abilities with all the power you have!" then yeah, barbarian would have to do his best to go nuclear on the bard.
But the other side of the medal is that the barbarian could say: "ok, so I just throw a rock at the bard, not even putting all my strength in it, too, for a whopping 1 damage! There, I have fulfilled the condition for the command!"
Obviously that would be a little bit of cheaty "interpretation" on the part of the player.
So, as per RAW, there always will need to have some DM call being made for such commands.
For example, if the DM determines that the charm is essentially word-based, then a literal interpretation of the command by the barbarian would work ok. Throw that puny rock! But if the DM determines that the charm is more like some kind of mind-control effect instead, then yeah it figures that the barbarian would attack with all his ferocity: he'd be obeying the actual intent of the Cambion, then, not just his words.
Now, even with 100% mind control, the mind control can be either Cambion-knowledge dependent, or not. If the command is Cambion-knowledge dependent, then it depends on what the Cambion actually knows or can guess about the barbarian. If the Cambion does not know that the barbarian can Rage, then the barbarian can get away with just doing the most ferocious "normal" attack can.
If the command is instead mentally all-pervasive, like actually temporarily turning the barbarian super-evil with only one goal in mind: "kill the bard ASAP!", then yeah even if the Cambion doesn't know that the currently battleaxe holding barbarian also has hidden in his backpack a super-costly magical bomb able to deal 9999 dg + save vs Disintegrate DC 30 vs a single target, that he keeps "for a really bad fight", the barbarian would not only rage, but make one less attack with his battleaxe, in order to draw the infamous bomb and proceed to really finish off the bard with the bomb.
So it really greatly depends on the DM's ruling of how "strong" that particular enchantment is here.
Since the DM just made the rolepaying description of the command quite boring and simple, only 3 words "Attack the bard!", I would have just forced the barbarian to attack like he normally does in fights. Forcing an enchanted PC to go all nuclear on an ally, require a bit more "dread and oomph" from the verbal description, like the DM saying this around the table:
Instead of simply:
"The Cambion then says: 'Attack the bard!'"
It could have been this description instead:
"The Cambion then says: 'Attack the bard!'. Suddenly the barbarian feels his mind reeling and stopping being his own. This is not a simple charm spell enchantment here, but a demonic-powered one! Everybody easily sees that the barbarian's face is now filled with extreme hatred, his spirit now controlled by pure evil. He now hates the bard with all his guts and obviously has only one goal: attack the bard, in the most powerful and violent way possible!"
The players would have had an intense situation, without the lash-back of ending up thinking that the extreme attack was too "DM-forced", but instead the natural result of the way that charm seemed to work.
But since the DM specified that the barbarian must do the attack to the best of his power, then that is what happens and the player really had no choice, despite the loss of agency.