Rear wheel losing alignment

I’ve got an issue where my rear wheel is coming out of alignment. I can get it aligned just fine when seating it on the floor. When I close the skewer, everything looks good. I can bounce the bike up and down on the floor, free-crank it off the floor, and everything is still fine.

Then if I grab the brakes and put my weight on a pedal, I can watch the wheel snap back out of alignment.

I tried replacing the skewer with one from another bike, same result.

I do hear a lot crackling coming from the hanger (I believe), when I push and pull on it, and even when I close the skewer. I don’t hear the same noise if I push and pull on the frame just above it. I can’t see the hanger move, nor can I visibly see that it’s bent, but I wonder if that could be part of the issue.

–UPDATE–

I swapped out with my road bike wheel and found that I cannot reproduce the issue with that one. Must be something with the hub or cups or whatever. Hopefully the bike shop can spot it.