the .22lr will work just fine. I don't shoot foxes around the place because they keep the deer mice and pack rat populations somewhat in control. But, I wouldn't hesitate to kill one that was raiding my chicken house.
The only issue I would have with calling foxes in AK would be the potential of calling in something a little bigger wanting an easy meal. I think a wolf, wolverine, or bear ( of either variety) would definitely cause some skidmarks inside your drawers.
I have had bobcats and badgers and even a mountain lion come to an electronic dying rabbit call intended for 'yotes.