HHKB obviously wouldn't be good for gaming as it doesn't have enough keys.
For FPS, it doesn't matter what board you use. the game mechanic is not highly dependent on strafe accuracy. and there's no apm spamming.
For RTS, Topre is definitely a No-Go because Cherry style linear switches are much more easily apm spammable
So in the end, if you needed 1 board to do everything, you wouldn't use Topre.
If you're gonna have 2 boards plugged in.. Then that solves the issue, you'd just swap when changing tasks.