Oh. For some reason I was under the impression that the adapter plugs into the USB port on the car.
Yes that's correct. However, when I plug the phone into that same port with a cable (instead of plugging the adapter into that port), then CarPlay works fine and doesn't hang.
The adapter is supposed to be replacement for that cable, and, to the car, it's supposed to look exactly like a CarPlay device plugged in with a cable. So if the car works fine with the cabled version of CarPlay, then it's the adapter's job to make it work exactly the same way. If the adapter hangs, then that's the thing that's failing at that job.
I won't completely rule out that the car's firmware isn't part of the problem. For example, I could imagine that the car firmware might be sensitive to timing, and the adapter might not respond to all signals with the same speed as the cabled version would respond. In that case maybe the car firmware might hang, but it's still the adapter's fault for not responding. Still, the other aspect of this car's infotainment system is certainly not bug-free, so I wouldn't put it past it to have some CarPlay bugs too.