All of the Free Hand Fighter require that they use a one-handed weapon and be carrying nothing in their other hand.
Now if you have claws you get a secondary attack with that hand that is carrying nothing.
Seems by RAW this works. Is there anything (other than anti-cheese) that stops this?
As soon as you devote that natural weapon to the attack sequence, it's no longer carrying nothing. It's wielding a natural weapon. So the bonuses of the Archetype would be suspended during the full-attack action (but gained back as soon as the attack with that limb is over).
Now, if you had extra arms from the Alchemist discovery, this would be a perfect use for them. Attack with sword, claw, and use that third arm to do your balancing, distracting, and other actions required of the Archetype.