Good news for all linux users: the game works without any problems with Fedora 39 and wine. Although a native linux version would be great, I can live with that :-)...

Which version of wine do you have?
I have a black screen after the editor and dev screen.

Sorry it does not work for you - for me it works with wine
wine --version
wine-9.1 (Staging)
out of the box... just after installation I go to my gaming users home directory with
cd
cd .wine/drive_c/GOG Games/DREDGE
and then enter
wine DREDGE.exe
that's it.

If this does not help you might also try if the game maybe starts with the heroic launcher - I had some good experience with that tool in the past. You can get it from here
https://heroicgameslauncher.com/downloads

I installed the rpm (for Fedora 39) as root user with
rpm -ivh heroic-2.15.2.x86_64.rpm
and then - as normal user - started the heroic launcher (either from the games menu or just by entering 'heroic' from the command line). Now in the heroic application just add the new game by entering a name for it and then the path to the executable (DREDGE.exe), and launch the game from heroic... good luck!

Thanks, my version of wine is the previous version, so it seems to be the problem. Thanks again for the info on Heroic :)