Meanwhile, I'm done testing with this setup. Because.. I accidentally fed +12V onto ... the USB interface inside my Nexstar+ hand-controller. The controller works just fine otherwise, but it no longer works with a USB cable.
It looks very likely to be just the USB-Serial chip that is damaged.
The PL2303SA USB-Serial chip still generates 3.3V power from USB, and the HC lights up when powered from USB. So it partially works. But it doesn't show up on the USB bus as a device. So hopefully it is just that chip. Easy enough to replace it. Those are about $1/each from China, so I've ordered a pair for CAD$2 including postage. I don't seem to have any devices on hand with one of those chips to cannibalize, so it'll have to wait 3-8 weeks for the replacement chips to arrive.
The HC continues to work just fine with the mount, so no problems for me other than being unable to do any more with this project for now.
And to repeat, it's not the Bluetooth project that killed it, but rather just me hooking up +12V to the 3.3V line of the ESP32 while the HC was connected to it. Oooops!