good question, most people use a tv track. this might be a better question for the GS remote forum, since its really a live performance question. better yet, psw has a live audio forum and archives.
here is a total shot in the dark though. guessing that since the tc voice doubler uses a foot pedal, that you could get a tech to custom mod a shure sm58s (which i believe has a SPST on-off switch built into the mic) disabl/bypass that and connect to a thin parallel cable and patch into the 1/4 pedal input of the TC, so you basically have modded the mic switch to be a single action footswitch working the TC.
Or if you have another mic of choice just wire your switch of choice into the 1/4 pedal jack and connect somehow to the mic, maybe even hold it in your other hand or connect it to your body part of choice.
something rigged like that might be worth a try anyways.