I still don't know exactly what my issue is, but I do know how to mitigate it. Apparently the randomly generated map that the server creates doesn't play nice with this mod pack. Using the default Navezgane map works with the 2 fundamental mods (SCore and XNPCCore) so now I'm going to try adding other mods and see what happens.
Randomly generated maps work with this mod on my computer, just not the server. Does that help you figure out what's going on?