I have this same problem and although you can source plenty of <compatible> Rolex crowns, as well as for other brands, no way to find any for the AP.
Only genuine articles that costs like a whole replica.
I suspect these replica crowns are very badly manufactured: the inner cylinder where the stem is screwed in, the one supported by a spring that should just slide out when you unscrew the crown from the case, is supposed to be blocked inside the outer cylinder at a certain height. The outer cylinder has a conical edge to grab the sliding one pushed by the spring and blocking it: in my case I was left with the crown between my fingers as the outer cylinder gave way to the sliding inner cylinder that remained screwed on the stem. There is no way you can reposition these two inside one another since they are assembled in reverse mode anyway (from up down without the actual crown, and then placed inside it) and besides that outer conical lip once enlarged or deformed I deem it cannot be repaired...
