I disabled the phone connection (not the notifications connection) to my Amazfit watch. There are two Bluetooth connections on the watch. Since then I have been able to connect reliably to the car kit. I still need to do a Bluetooth restart from the Utilities option in Settings to get the phone to connect sometimes (usually after a phone restart or after Flight Mode has been in use), but otherwise it works OK.
I suspect my problem was the audio was being routed to the watch even though the car kit initiated the call.