Since FMV songs are orchestral and almost all of them appear in only FFIX OST Plus, it's very hard to find midis for them. I don't think the FMV with "Breaking Through South Gate" had any background sound effect, only the music, so if you want the song without buying FFIX OST Plus or downloading the mp3, you can just extract or record it from the FMV. The programs mentioned by Silent Warrior should be able to extract the song.