Hi Sue,
I think the idea is a lovely one, and Susan is right, it is backed by the hide. That being said, there is a needle designed specifically for hand stitching of leather. It's called a Glover and I have one that came on a card of special needles by Coats and distributed by Dritz. The entire card is called Homecraft Needles.

I think the actual leather from a fur coat is probably thinner and finer than most leather things, more like light weight glove leather, which can certainly be stitched by hand. You'll want to keep the pattern a simple one, though. Especially because of the multiple pelts, as Susan said.

Actually, Susan's Bunny Doll Pattern would be a good one to try, but I'll look around for a bear pattern for you.

I hope this helps.