They make trackball clickers[1] (often with a laser since they're for presentations), and other single handed devices, a search for "finger mouse" surfaces a few different devices

I don't know if any of them have PTT with microphone but you should be able to pair the clicker with a regular bluetooth headset, while the headset may be on all the time you might be able to use something like joy-2-key and a macro/hotkey utility to map the button press to mic mute.

[1] https://www.amazon.com/Wireless-Handheld-Finger-Trackball-Po...