I do not know how to change the order of the stages. I also tried restoring gore to the Pit Bottom and Goro's Lair. Every edit I've tried crashes the game.
No, not the stages per se, just the BGM order. I worked on the UMK3 hack, and noticed a string of preset BGMs at round beginning, climax and end. I guess this is just one string only, since there's no stage transition.