I just fixed this problem for myself. Early Call of Duty games had the case-insensitive issues as well. COD4 is perfectly happy using the right system calls.
In at least enough instances to matter, Avernum 6 uses full pathnames of files, not relative pathnames. This means if you have a case-insensitive partition or DMG that you keep the game on, but it is mounted on a case-sensitive filesystem, it still won't work.
Here's my specific situation, and how I fixed it.
I keep case-sensitive games in a DMG .. call it "Games". It is normally mounted in the filesystem at
/Volumes/Games
Everything under Games is case-insensitive. The root filesystem is case-sensitive. So, when Avernum 6 tries to access something like
/volumes/games/avernum 6 ƒ/avernum 6 files/graphics/g1520.png
...it fails. As far as OSX is concerned, Volumes and Games needs to be capitalized. Everything after that is case-insensitive.
So, I have created a directory in the root of my filesystem called "/volumes". OSX knows this is different than the system-default "/Volumes".
I create a symbolic link under /volumes called "games". It points to /Volumes/Games.
That way, when the program tries the full path, all lower-case, it ends up in the right spot.
In a console/shell:
mkdir /volumes
ln -s /Volumes/Games /volumes/games