iPhone 13 + iOS 15 causes intermittent handsfree dropout

My vehicle does not have CarPlay, but does have onboard Bluetooth handsfree, and a separate Bluetooth audio input. So my iPhone connects to both Bluetooth devices, using the handsfree device for calls, Siri interaction, etc. and the audio device for streaming audio from iTunes, YouTube, etc.

When I had an iPhone XR, everything worked perfectly. Once I upgraded to an iPhone 13, though, the Bluetooth handsfree audio drops out intermittently as long as the secondary Bluetooth audio device is enabled. Phone calls are impossible, because I miss at least half of what the other person is saying. They can always hear me, so outgoing handsfree audio is OK. If I pull the plug on the secondary Bluetooth device, the handsfree clears up.

I've tried the standard troubleshooting, delete all Bluetooth devices, reboot, reconnect, etc. to no avail.