Okay so here's the deal. Most of the Final Mix tracks were released, this is true. However there is one track that was never released on any soundtrack, not even the Kingdom Hearts Complete. (TRUST ME, I've checked.) And it is played mostly during Sora's memories of he and Riku as heard here: Always On My Mind (Unreleased) - YouTube (http://www.youtube.com/watch?v=ISikigOpYwY)

Would anyone be willing to make a gamerip of this part so that way it wouldn't have those horrible sound effects interrupting the beautiful piano? Or does anyone know of a game rip that was already done and includes this track in it?

This track is on the Kingdom Hearts (1) OST.
It only has no loop.

No, the difference happens about 1:22 seconds in... Listen to both versions, you'll notice the difference.