Not to kick the Mutagenist while he's already down and bleeding, but I just noticed that the Unarmed Strikes you get from Bestial Mutagen lack the Unarmed trait. That's the trait that makes an Unarmed Strike not take up a hand. The general rules on Unarmed Strikes do not say that they don't need to be wielded, only the trait does that.
So Bestial Mutagen means no shield, no Quick Alchemy, no Quick Bomb Bottled Lightning to the face, no grappling.
The only small saving grace I could find was the fact that the mutagen gives you "a claw", so it might just turn one of your hands into a claw. That would make the situation slightly better as long as you don't plan on using a shield. But personally, I would totally want to use a shield as a mutagenist.
The Unarmed trait is also the only thing (by RAW) that makes those Strikes immune to Disarm. So you could totally Dis-Arm the claw from a Mutagenist.
In comparison all the Monk's Style Strikes and all Barbarian Animal Instict attacks have the unarmed trait. The special Unarmed Strikes of druid (Wild Morph) and sorcerer (Gluttun's Jaw, Dragon Claws) also lack it.
I assume it's an oversight - especially since wielding a Jaw attack doesn't make any sense.