I think I might know the problem. I had this problem at least. It might be the same.
This one site told me that the stage .pac file needed to be named STGSTARFOX_ASTEROID.PAC. That did not work. I wanted to test something, so I found a download that WAS for Lylat Cruise, and looked at its .PAC name. I named it STGSTARFOX_GDIFF.PAC and it worked. The module file, if you need one, is just named st_starfox.rel. That might be the problem. All I did was name it STGSTARFOX_GDIFF.PAC and it works fine. Hope that works for you.
And I downloaded the stage you linked and looked at it in brawlbox. I'll assume it's originally for Brinstar because, well that's what the maker said and also said the .rel included was to put it over Yoshi's Island (Melee). Assuming it IS originally for Brinstar, here's the module file you'll need to play it over Lylat Cruise (unless you have one).
st_starfox.relLike I said, unless you already have one, download that, put it in your module folder and just rename the stage .PAC to STGSTARFOX_GDIFF.PAC and it should work. At least that's what I have set up on my SD card. The stage file name threw me off, cause I saw several different ones. If not, you could try the other names it has, but this one hasn't failed me.
