Having looked around a bit on the web, it seems that the main problem with playing with seven players would be that there are not enough cards to build seven players decks. Would buying a second classes add-on deck remedy that or are the cards from there too class-specific to help out?
Playing with 7 would also create a problem with regard to locations. Each scenario has a certain number of locations based on the player count. I would think it might unbalance things too far towards the players if the ratio of locations to players was off.