Considering everything is already unlocked there isn't a need to save. Your "progress" that would be rounds you've beaten during that particular game play mode. If you are doing well enough to earn a record, but leave prematurely, that's the progress you would be losing.
Play until you've beaten Arcade, Time Attack, or lose in Survival- or don't. There isn't anything you are really going to "lose."
You can do it through as many sessions as possible. Just don't "quit" until your currently selected character has won. If you do, you'll have to start over with that character.