The LM386 has no inhibit control so we have to find different solutions, if possible. As Jerry reported "Dumping 12v into the bypass pin of the LM386 to quiet it down is a bit unorthodox". If you look at the internal schematic of the LM386 you will see that taking pin1 to +12V will block the amplification of signals to the output push-pull stage. The relay commutation should take a few tens of mSec to switch and the LM386 gain being zero, as the PTT is de-pressed, will block any pop up plus the RX +12V will quickly go to zero, NO audio output available. No interference with TX stages is expected.