Unless I’m reading melee.cpp wrong, bashing damage (and bashing skill) from the unarmed weapon is totally ignored, while cutting/piercing damage is added as if it wasn’t an unarmed weapon.
However if your weapon is an unarmed strike weapon, your claw mutations will add their cutting or piercing damage (they add to both, but only the higher value is used) as long as you have nothing on your hands.
Actually not nothing - for some reason mutations add cutting damage to attacks only if you ARE wearing something, while stabbing ones only if you aren’t. That’s probably a bug.