I tried Lutris with different runners, the system Wine, PlayOnLinux, adding non-Steam M2TW to use the Proton layer, to no avail. Most often, I just get a frozen black screen right after starting the game. After many many attempts and many many hours, I finally gave up my old CD version of M2TW and bought the Steam version of M2TW Definitive Edition (no sale for Labor day???).
I copied the rotk/ subdirectory from my old Windows machine to
/home/yourname/.local/share/Steam/steamapps/common/Medieval II Total War/mods/
on my Linux machine.
I ran the game under Proton 4.11-13 with "--features.mod=mods/rotk" in the Launch Options, and voila!! At last! Played a few turns and fought a battle already. Smooth sailing so far .
Note that's the native Steam of the Manjaro distribution. Running the mod with Steam Linux runtime gave CTDs right after launch (the vanilla game runs fine).
When I used Proton version >= 5, I kept getting CTD when starting tactical battles. I also got vertical black stripes on the strategic map (around the tiles?). With Proton 6.3 and experimental, it's even worse. The fonts are all weird and unreadable in the Options menu, the in-game messages, and for the character names. The CTD and black stripes happened in the vanilla game too, with Proton >= 5, not just the mod.
Hope this helps...