I would allow a barbarian to select a fighting style instead of reckless attack.
I wouldn't. Modifying the rules to permit a player to take a trap option seems like a bad idea.
I mean, I might, but I'd also want to give the barbarian something else.
Having said that, I would allow a Champion Fighter to choose reckless attack as their second fighting style.
The Champion fighter has problems long before that in my experience.
In fact, by around that level, the Champion fighter is starting to close the gap with the alternatives. A Champion with reckless attack, 4 attacks, and 18-20 crit range is going to be a beast at level 20. It might be a little too good.
It definitely overshadows every other choice at that level.
I mean, at level 10, the Champion could probably get every single fighting style in the game and it wouldn't really break things. That would be weaker than Reckless Attack.
Fighting Style Mastery (Alternative subclass feature)
At 10th level, once on her turn the Champion can change her fighting style. In addition, the Champion can draw a weapon as part of an attack with it, and sheath a weapon before she makes an attack.