Some of us on the BMG forums have discussed some ideas around campaign play, but it's really tough to do in my opinion. You simply can't allow characters like Batman to gain many more tricks or stat increases than what is on the card before ending up with crazy powerful fiends.
The scenarios provide some nice variety and it's pretty easy to come up with your own inspiration. Based on my experience so far I don't think the lack of campaign play for BMG is actually that important.
Some ideas to use instead of beefing up characters,
-Grow the stable of characters that can be used to play with. Players start off with 1 characters (or, if you're playing police, Gordon and minimal points of generic cops). As you play, you can purchase additional allies. So Batman can eventually get Robin, which can then upgrade to Nightwing or he can bring in Batgirl.
-Have a "Bat Cave" or secret base which give him access to single use bonuses rolled from a table at the start of the game. Higher levels can get more bonuses per game. These should be costly, or limited as to how many can be purchased per advancment.
I could easily see a league being set up with this where players each take on a character, then after that, any unselected characters can be purchased as allies as players level.