This post might be useful to you. It is code for gettng the "hidden" F13-F24 keys working with a due or pro micro (others HID compatible boards will be easy to adapt - those are just the boards I happen to have)
You could then bind "F13" to PTT within TS3, and "F14" to mute/enable mic etc.
https://forum.arduino.cc/index.php?topic=324554.msg3725988#msg3725988
I built mine using "guitar pedal" momentary stomp switches and have the keys bound within Open Broadcaster Studio for scene changes etc - no reason the same kind of tool wouldn't do what you need.