Right, I confirmed UnityModManager looks different when I change the GTK theme.
Then, instead of Kate, I tried changing the GTK theme like you showed in Lutris and that works.
So the issue seems to lie with Mono. I found this old issue https://github.com/mono/mono/issues/9998
Mono seems to ignore the GTK_THEME environment variable. However there is a workaround: GTK_DATA_PREFIX= mono ./UnityModManager.exe makes mono default to a classic window theme.
Thanks for your help!