Jonathan - you are no doubt feeling frustrated. This is one of those situations where sitting back and taking a deep breath helps. This is likely a simple troubleshooting issue.
Start at the circuit breaker that powers the EFIS. Connect the black lead of the voltmeter to a solid ground and clip it there. Using the red probe, place it on the bus bar that feeds the hot side of the circuit breakers - measure the voltage. Now do the same on the hot side of the CB (not the screw but on the CB itself) and again on the switched side of the CB. Now do the same thing as you key the bad VHF comm.
I'm thinking your EFIS CB has a screw loose or has become somewhat resistive. Keying the VHF is causing the bus voltage to drop. There's more than a small chance this kind of "loose connection" exists elsewhere, like where the circuit breaker bus bar is getting its power from the aircraft battery.
Just to provide an example of how this works, I recently discovered a CB that was introducing more than a 1 volt drop. The CB hadn't failed, it wasn't popping or interrupting current flow, it was just getting warm because its internal resistance had gone up, thus causing it to drop voltage.
I wish you luck in your investigation. Post back here with findings so we can help get you to the source of the problem.
