Hmm. I'm going to guess this is possibly water-hammer related, where a high-pressure spike caused by the toilet fill valve shutting is forcing water past the mixing valve. If so, the water would be released as the toilet stops filling, and a water hammer arrestor should settle it down, be it new and high-tech with a piston or old (merely an air-filled pipe stub that needs to be drained of accumulated water from time to time.)
If it releases water at the start of the flush, I'd have to suspect some sort of vacuum breaker being involved, and getting activated by the sudden drop in pressure when the toilet fill valve opens.
Edit: ahh, your new information is possibly diagnostic - turning the hot water supply off stopping it suggests that the casue is probably related to the "anti-scald" function that's pretty much universal in tub/shower valves at this point, which is responding poorly to the pressure drop on the cold line as the toilet fills (and for some reason, doing so even if the mixer is shut off.) You might call the valve manufacturer and inquire if this is expected behavior or a defect of some kind.
If it is expected behavior, and you cannot run a new line that's either larger (for less pressure drop_ or dedicated (so the toilet and shower are not sharing the line) a small pressure tank/accumulator in the bathroom area should sort the issue by holding pressure up while the toilet fills.
On third thought, first go though and make sure that all the valves in this line are FULLY open. If the problem persists, see if your toilet fill can be adjusted to be slower. Then consider more involved solutions.