Ah! I don't think you'll be able to reproduce that in a music editing software, I'm afraid! Because of the way the Gameboy processes its sound and music, it can't be manipulated in a sound file.
The Gameboy's sound works by using a bunch of individual sound channels, and playing them at once to make the music and sounds you hear in the game. Think of it as a music group and its members. If you're at their band rehearsal, and you want to change only how the guitar sounds, you can tell the guitar player to play differently. But if you have an MP3 of one of their songs, you can't tell the guitar player play differently, because it's just a recording of what they played.
The same goes for how the Gameboy's sound works. Lameboy's speedy-uppy music quirks can't be replicated in sound-editing software, only when you're actually playing the game. In Lameboy, the game "commands" how each of its "channels" are to sound, the same way you can "command" band members to change how to play. When you speed up the game in Lameboy, it sort of messes up the commands due to emulation quirks, leaving some of the channels sounding borked when you resume normal gameplay speed (at least until the Gameboy issues another command, in which case the channel may fix itself).
The same thing can't happen in an, say, an MP3 of the song in Goldwave or Audacity, because it's not an actual Gameboy and each of its individual channels, but just a recording of all of the sound it makes, mixed down to a single channel. Because of this, you're only able to change the pitch of the entire piece in music-editing software.
Sorry if it doesn't make sense! Explaining this sort of thing is really tricky business. Maybe someone else can explain it better?!