Linux isn't officially supported.

I know that :)
But many games aren't supported on Linux, though run on Linux (using Proton, Wine, etc.)

Works perfectly if you use dxvk.
May require Wine with gstreamer support compiled in and x265 support for videos.
Besides that it ran crash free over 1 day for me so far.

tldr:
vi ~/.bashrc
add alias biomutantgog='export WINEARCH=win64 export WINEPREFIX=/home/wine/biomutant; cd /home/wine/biomutant/drive_c/GOG*/Biomutant'

then
$: biomutantgog
$: winetricks dxvk
$: wine setup_biomutant.exe


edit: there seems to be a issue with Media Foundation Framework.
If you just get a black screen, move the intro movies under GOG Games/Biomutant/Biomutant/Content/Movies somewhere else.. Also projectors in game do not function without mf installed. Some freezes may occur with media foundation framework. Without the intro movies i can start the game by pressing ESC a bunch on start.
Using Wine 6.9 with dxvk, media foundation framework is from here, but the game may be buggy if installed. https://github.com/z0z0z/mf-install
Seems like mf is a curse of Unreal Engine 4.