Because sometimes when I'm playing on Project M, the normal Brawl character and stage screen appear.
Yeah that happens to me too. Only after the music gets all garbled or whatever, and I quickly reset the match. It goes back to Brawl stages, music and physics. Idk if that's what you mean there.
About what you're asking before, I don't even know how that's possible. The only way that could happen normally is if you changed all those song settings and kept forgetting lol. But maybe you could try going to the wii saves menu and copy brawl's data? That's what I would try, for the sake of trying. Though I don't think it'll help, and I honestly don't think anyone but Nintendo could address this.
If the issue is only on Project M and not Brawl, then yeah it could possibly be the SD card. Do a format in that case, like they said above.
After that, if it still didn't work, your other 4 options- call Nintendo, deal with it, turn off the music altogether (what I'd do), or last resort- do a system format. Everything from Brawl would be gone in that case, but I was told that Project M unlocks everything.