I am being forced to always have "Krastorio THE FACTORY MUST GROW" as the background when I want to play on my friend's server. If my mod startup setting match save for the "Main Menu Background" selection, I get the "Your mod startup setting do not match" message and have to restart the Game just because I ned to load it with that specific background instead of the one I want.

Would it be possible to move the selection to another area and still have it apply, the "Per player" menu perhaps?

No is not possible because Factorio developers have choose that game setting as "player setting" or "game setting" are loaded only when a game is running, the only setting available in the main-menu are the "statup setting", and also for how the game is designed they must be shared.
If u really want this u can ask to the developers here:
https://forums.factorio.com/viewforum.php?f=28
But I'm pretty sure of the answer (because I already ask this question in the past).