What would happen if e.g. Priest and Paladin shared a "class card"? Both are adepts of the Light, seems logical they could share some skills here and there, even though their fighting style is different. Instead of neutral cards where all classes can get it, only 2 classes can get this "shared card".
For example only Paladin and Priest can use this:
Or they could share Holy Light and probably a few more existing cards that is not game-breaking.
A while ago, I had a concept to make 'merged' classes. They would be new Heroes with new Hero Powers, but would be able to use two Classes class cards. An example would be like Tyrande, as a mix between like a Priest and a Druid. This would be difficult to implement though as there could be some insane combos and synergies between classes, so there would have to be limitations, I guess.
I can't see any problem with the idea in theory. Obviously you would still only be playing as one class so it wouldn't change the game massively, it would just allow a release that gave more options for each class (because each class card would serve double duty).