So I just install that and the 1998 versions should work fine?
Extract it into the game directory where the main executable is, yes. No "installation" really, worst case re-install the game and everything is as before.
As I haven't heard of your particular issue I can' t promise it will help, but it has helped people with quite a few issues in the past. Plus you actually get full-resolution 3D hardware rendering, which the standard GoG versions do not support.
That last part also applies to XvT/BoP (note that BoP binaries are in a separate directory, so you need to copy the files to both places in case you want to try it for that as well)..
It is the same solution (well, with quite some additional improvements) that GoG uses to make X-Wing Alliance work on modern computers
Dropping that file into the game directory did make both X-Wing and TIE Fighter appear normally, but the mouse cursor is invisible. It's clearly there since I can still highlight and choose options if I happen to roll over them, but I can't tell exactly where the cursor is on the screen.