I’ll preface this by stating that I do realise that by RAW natural weapons are not compatible with two-weapon fighting.
However, would it significantly unbalance things or break anything to allow it? The PC in question is a Barbarian of Path of the Beast and they get a bunch of natural weapons options at level 3, specifically they want to use their claws for it:
Claws. Each of your hands transforms into a claw, which you can use as a weapon if it’s empty. It deals 1d6 slashing damage on a hit. Once on each of your turns when you attack with a claw using the Attack action, you can make one additional claw attack as part of the same action.
The 1d6 damage that the claws deal is on par with handaxes, scimitars and shortswords, all of which qualify for two-weapon fighting so I don’t think allowing it is going to make the PC dramatically more powerful, after all they could just grab a pair of handaxes and have the exactly same damage output. I do realise that with the claws they’ll get one extra attack as part of their Action, which they would not get with weapons and I’m ok giving it to them.
Also the PC is level 5 already and have the Extra Attack feature so technically they could just attack with their claws first to get the additional attack and then switch to the axe to still qualify for the off hand bonus attack.
Is there anything that I’m missing or forgetting about this that could cause problems down the line?