Oh, Kim. I think you may be in trouble with this one. Is this thing bear sized? It's going to be huge and heavy - and need a very deep shadowbox.
I suppose the customer is bound and determined that is HAS to be framed, so I should stop being so negative and try to be helpful instead.
If you can't find a conservator, you might check with someone who sells/restores fur coats or leather goods. There may be some way to make the skin more supple. Without disturbing the painting of course.
Plan a frame with enough depth that you can fit the backing board slanted away from the bottom of the frame so the skin doesn't have to support all of its own weight.
Line the shadow box with whatever color the fur is - it's gonna shed.
Keep in mind that plexi is going to be static-y and might make the fur stand up - giving the whole piece a frightened look.
If you can't sew it, perhaps you can find some spots where you can thread leather thongs through it and tie it to the backing board.
I would be tempted to explain the mounting problems and hand it back to the customer. Let him be responsible to getting it into good enough condition to be framed.
Good luck.
Kit