SIGTSTP is canonically the signal for suspending a foreground process, and (almost) the same as
ctrl+z. The obvious benefit is that this halts its use of resources and any other impact it may have; it freezes the state of the process and yields control to the parent process. The less obvious benefit is the TTY component: you immediately regain control over stdin/out/err, and when you resume the task, you either retain control of execution and I/O with
bg, as if you were running it like
prog &, or release it with
prog.) I know that SIGCONT will resume the process, but I don't know if or how it decides between FG/BG.
Now, the difference between SIGTSTP and SIGSTOP is that SIGSTOP is non-negotiable, whereas SIGTSTP can be "trapped" (handled/caught/etc.) Here is a simple example and here are a few caveats. Try
man 7 signal - you may find another collision, if not a way to use them to your benefit.
There are some 'exterior' commands that perform the same role - you may be able to accomplish this with
ixoff, but I'm not an expert and I haven't done much of anything with signals. Maybe keep a cheat sheet for when you start testing, because it seems like it would be very easy to lock up your environment.
Edit - it wasn't fully clear from your post - would disabling the default Tmux binding solve your issue?