Forget about mono and MBII Launcher, they currently aren't going to work on Linux.
The fact you ever getting kicked with "... keep Launcher running" message means you're running Windows version of MBII via emulator.
Forget about any attempts of using any emulators like Wine, Proton, Cedega etc. They aren't going to work.
MBII Launcher has built-in anticheat, but that anticheat is only required for Windows, not Linux
There's native Linux version of MBII. You must use it. All required files (highly likely) are on your disk already.
Start file mbii.i386 with this command line parameters:
Do it either by typing command directly in terminal, or edit file play_mbii.sh (replace openjk.i386 with mbii.i386).
Be ready that you'll need to set up multiarch.
Absolutely majority of operation system these days are 64-bit, but Movie Battles is strictly 32-bit game.
64-bit Windows handle 32-bit programs out of box, completely automatically, but for Linux it's necessary to put some effort. Nothing really to hard, though.
During first launch the way I described, MBII will most likely crash, and will complain about missing dependencies.
Read crash message carefully and install all libraries Movie Battles will cry about.
Install ONLY 32-bit versions of those libraries using your package manager.