Not that I can likely afford a polar bear hunt, and not sure I've really got a huge interest in hunting them, but...

Is there a Canadian market for the hide? Couldn't a fellow enjoy the hunt, return home to the U.S. with the requisite photos but no actual bear? Sell the hide to someone in Canada so that it doesn't go to waste?

Why not?

OK, there's no big white fluffy rug in the U.S. house/office, but there's the satisfaction of having hunted the great white bear, and photos to prove it happened. I'd be content with that.

Guy