You could argue that claws are Unarmed Strikes, since you aren't arming yourself with a manufactured weapon, but instead are using the weapons you were born with... the weapons God gave you from birth to prove your, and therefore THEIR, superiority!
Beltar even favors the spiked gauntlet, which is just a puny human's way of emulating the claws you were BORN with!
Those deity lists were written by puny humanoids with weak fingers where their claws would be if they were born strong. That's why they write Unarmed Strike when they mean Natural Weapons. They figure animals don't worship gods. Well they're wrong! Some animals ARE Gods! Take the Dire Tyrannosaurus, or the Kaiju Hamster, or the Weasel of Legend.
Never heard of them? That's cause they're waiting under your bed.