The hat in the Bodmer painting, to me, looks like a pelt that may have had the belly and legs trimmed off to make sort of a rectangle. It could then be folded in half and stitched up one side. This would form the peak or point on the top of the cap in back, and would be very simple to make. However, if the face and tail were left on, you would see one or the other hanging over the man's shoulder. I think I see the badger's face over the man's right shoulder in the painting. The wardrobe people for the movie evidently left the face on, too, and somehow manipulated the skin to put the face in front. So, the movie hat looks a little different, but I do think it is of badger fur.
Alfred Jacob Miller was an artist who actually attended one of the Rocky Mountain Rendezvous and painted the trappers and Indians from life. His paintings are excellent resources, as are those of Karl Bodmer, who traveled up the Missouri in the 1830's. There was a 20th century artist and researcher named Rex Norman whose interests have moved on at this point, but 15-20 years ago he was immersed in the Rocky Mountain fur trade culture. He wrote and illustrated a number of excellent articles for MUZZLELOADER magazine at that time, and he also authored and illustrated a book that might interest you:
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This is the page on fur caps:
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The book is a study of Miller's artwork, with explanatory notes. this book would be considered a "secondary reference" by hardcore reenactors, but Norman's research is impeccable, and I would consider it "trustworthy."
So, I can't say exactly how the fur caps in the painting or the movie were made. I made my best guess for the cap in the Bodmer painting, but the movie cap seems kind of formless, maybe just pulled roughly into shape and held there with thongs. Hard to say. It looks warm, though!
Notchy Bob