After attempting all the Lutris scripts available on the Lutris install webpage, consulting the outdated Renegade X Linux guide and having no success at all (Lutris errors out at the start of the install, with an unhelpful error 256 or error 13568) I slept on the issue, then decided to try something else.
I had kept my Renegade X folder from a Windows install, which I dragged over into my Home/Games folder on Linux Mint, then I went to Lutris and added the folder as "Add locally installed game".
Name -> Renegade X
Runner -> Wine
Then
Right click on Renegade X in the main Lutris window, chose Configure -> Game options tab.
Executable -> Renegade X Launcher.exe
Working directory -> Renegade X
Then under Runner options tab
Wine version -> lutris-GE-Proton8-15-x86_64
This last bit might only show "System(versionnumber)" and "lutris-(versionnumber)" for you. I had added GE-Proton8 by using the software "ProtonUp-Qt" available from Flathub since before. I do not know if this is required, however.
This uses the classic Launcher.
Sidenote: If you want to run Renegade X through Steam, you should use the Steam client available for download from store.steampowered.com and not the Flatpak, otherwise it will not have permissions to view other installed software outside of Steam itself.
Remaining issues now is while I have added Renegade X to be launched through Steam as a non-Steam game, Steam does not seem to display the game as "Running" and thus never enables the Steam In-Game UI. If anyone has any ideas how to solve this, please write a post about it!
The other issue is that the Launcher (not in game) seems to forget all settings between each startup (username, skip intro movies). Not sure why this is the case.
Excuse me if something is I wrote is incredibly dumb or obvious, I am not a Linux expert.