As far as I can tell, upgrading your level has no effect on the game apart from changing the music and your trail, does it affect your score and do I need it to get the bonus stages?
Basically, the "extra" stage is just the highest multiplier you can get for your score and it doesn't affect whether you get the retro bonus challenge at the end of the level.
Getting this multiplier is part of what you have to do to get "Perfect" on a stage:
In the Steam version of the game, you also get an achievement for reaching Extra and for getting PERFECT!! on a challenge.
Upgrading your level gives you points and increases your multiplier.
If you don't upgrade your multiplier, you won't be able to achieve a "Perfect!!" on that level.
(And you'll immediately fail the level if you're playing on Perfect difficulty).