Small update on this after doing some more tinkering. Earlier I was using the nightly Ikemen version which isn't a full release, so if you just use the old 0.99 release all you need to do is copy over the executable script to the crushing force game folder and you'd be good to go.
Although those 3 stages still won't load, after doing some digging, it seems like it has to do with the Linux port only using OpenGL (As evidenced by the select-GLOnly.def file in cfhud being the same as the select.def file but with every instance of those 3 stages replaced), haven't found a workaround for this yet, and it probably won't be fixed until a new version rolls around that has better support for OpenGL or other, more modern renderers.
Also proton does work fine, this is just if you don't feel like setting it up or just want to use the native version for some reason.