There has been a moratorium on Polar Bear hunting since 1973.
No hunting from aircraft or motor vessels and in some regions only Inuit can hunt bears.
As for shooting a Polar Bear with a .22, a .22 caliber would bounce off a bears noggin like a chick pea and wouldn't even penetrate the fur and hide let alone four inches of fat.
The Inuit hunt with .303s even then Polar Bears are extremely dangerous prey.
I've seen one running away at 40 miles an hour after being wounded and it took half an hour to take it down from a helicopter even then it reared and took a swipe at the helicopter.
Preferably with said rifle functioning flawlessly.