I will try that fix for the audio amp noise burst by muting the LM386 via pin7.
Seems that there are two problems here.
One is the audio noise burst which your fix addresses.
The other is a burst of RF noise as charging capacitors in the microphone amp unbalance the balanced mixer at the start of a transmission.
I have looked through the Wiki and there are lots of discussions about curing the audio burst, there is not much about fixing the RF noise burst.
I have been thinking about putting a FET in the line from the mic amp to the balanced mixer to act as an isolating switch while the mic amp settles down. Use an RC time constant on the gate to delay turn on. Use some steering diodes so that the fet is on via the Rx line during receive, goes off when the Rx/Tx changeover occurs and comes on after a delay via the Tx line when in transmit.