Toward the end, the manitou's claws shred a shower curtain. In the next shot, its hands show no claws.
Native Americans have never used funeral pyres as part of their death ceremonies. That practice was limited to the Old World, specifically Europe and South Asia.
In the funeral pyre scene, when the man touches the flaming torch to the pyre, there is a significant delay before it ignites, and from a totally different location than where he placed the torch.
When the transforming manitou shreds the shower curtain, the tears start at the fingertips not the claws.
When Lyle is in the bathroom turning into a werewolf, he lets out a big roar. Scully is right outside of the door trying to pick the lock and yet somehow she does not hear this and keeps trying to pick the lock.
In one scene, you can clearly see that a US flag patch sewn onto the right arm of the Sheriff's shirt has the star field on the left. This is not how flag patches are supposed to be shown. A little later, you see the Sheriff's jacket has a flag patch sewn on the correct way. As a matter of fact, you can see the incorrect version on the photo page for this episode here on IMDB.
If the shapeshifters shed their skin when changing between human and manitou and back again, then there should be manitou skin when it dies and changes back to human.