Logitech P710e conferencing speaker+mic shows up as "Headphones"
I have a Logitech P710e here that works fine as both speaker and microphone when connected via USB, however when connected to a Fedora laptop in Bluetooth through GNOME Control Center, it sees it as a "Headphones" device (in the Bluetooth panel's device properties) and in the Volume control panel only the speaker shows up, not the microphone input device. Comparatively, pairing it with an Android phone, the P710e's microphone gets correctly detected and used (I tested with an echo test and the hardware's mute button).
I don't know what information to provide to properly debug this, it seems all I have is the device's "address" (B8:D5:0B:F4:9A:F6) but I doubt that's useful? And if I plug the device via USB, here's what I get with lsusb:
Bus 001 Device 067: ID 046d:0a4e Logitech, Inc.