Wiring – Why doe the AFCI immediately trip

Given that you state that "no power flows to the room" when you use a normal breaker, the most likely explanation is either that there is a short/arc somewhere, or the wiring is incorrect.

Shorts/arcs are not something to be messing around with. I would call in a professional.

For everyone's info, and because it could help someone else, here is the actual answer in my case.

While I had connected the white wire coming out of the AFCI to the ground bar, I hadn't connected the neutral from the circuit to the AFCI. In my case it was because I didn't realize the extra screw terminal was actually there and the documentation wasn't helpful.

Bottom line: If you install a combination AFCI and connect the hot but not the neutral, it will not pop until there is a load. After the fact, that actually makes sense. This is like a ground fault from the AFCI's point of view because it can't see the current going back through its neutral port.