With help from /u/duynguyenle, I found out why my Anne Pro won’t work wirelessly: the battery had been punctured by the switch solder joints underneath the PCB.
Aside from his and my units, I haven’t seen this problem reported anywhere else. Still, it might be worth checking; this is a huge fire hazard and should be dealt with immediately
Edit:
I replaced the battery with one from my KC71 and now it works wirelessly. I’ve also snipped the switch pins directly above the battery to be flush with the PCB, and added extra foam for additional protection. Because the replacement battery is physically larger than the stock one, I had to dremel out some of the plastic struts inside the case to make it fit.