Just a quick update on the hands-free solution.
The stock earpiece-microphone of the Baofeng UV-82 is quite lousy with its muffled sound and insensitive microphone. At least that's how my specimen behaves. So it is not a big deal to sacrifice it.
I removed the microphone capsule and used it to feed the audio coming out of the wireless receiver. I added a 470nF capacitor in series to block DC. Initially I chose 10nF but the resulting signal was too low (checked on the oscilloscope). Also remember that this C makes a high-pass network together with the input Resistance, so your voice could be attenuated further.
The first on-air test resulted in low audio. Then I hit twice the [+] button on the transmitter and the transmitted volume was back to normal. An unknowing HAM did not report a different sound of my voice, so the whole thing works.
VOX can work but it is insensitive and on UV-82 there is no setting for the hold time.
Next step will be to build everything into a comfortable box, add a lever PTT and possibly a way to power up the receiver together with the RTX. It has to work until I don't receive the external antenna and check if it can handle the RF power of the mobile transceiver I still own.