So after working on the problem for a while longer it turned out to be a combination of 2 things:
1) I needed to be using the Knossos launcher, NOT the FreespaceOpenInstaller.jar and wxLauncher.exe recommended by the installation guide in Steam. Everything just works SOOOOOOO much better with Knossos!
2) The game itself was crashing because of the MSI Afterburner in-game overlay
. Good call, @The E. As a side note, I had also been unable to get the new Call of Duty: Modern Warfare to run either, and it would always crash on launch. After disabling the overlay got FS Open to run, I opened CoD as well, and it suddenly started working!
So, lesson learned. Ingame overlay does not play nice with some games.