Both sound like really good ideas to me. While they're both a bit limiting, I think they'll make room for a lot more possibilities and give a new and fresh way to approach the dungeon. One of my personal gripes with the dungeon has been that, having participated in it 4 times (I think), I've just been using the same characters over and over. My RPG library is regrettably not as large as some of yours, so I'm a bit more limited in my choice of characters. I'd find myself just picking Eiko/Garnet/Red XIII/Sharanda over and over as my token reviver(s), and I think it did start feeling stale.
If I could just address a couple of things, the way you're thinking of operating Game-Centric Teams could be a little bizarre or imbalanced. Some casts may end up deceptively powerful (as mentioned with FF4, you've got 5 characters, which, regardless of character selection, is 2 extra bodies of HP that enemies need to take out compared to your typical 3-party game; that's a huge advantage.) Also, how would you handle games like Fire Emblem, where party size could be even more problematic? It's a good idea, I just think it could be fine-tuned a little.
As for the Class Challenge, that'd be really interesting. I know someone did a Final Fantasy Classes tournament not so long ago, and that was really fun to watch unfold. The only thing is that, maybe even more so than the Game-Centric Teams idea, you could find yourself in a lot of sticky situations. Several strong characters that would be worth having on a team don't have an obvious class (Is Tidus a Time Mage? What's Wakka? How 'bout characters like Aika, Edgar, Kratos/Zelos or Feena?). It feels like a Class Challenge system favours certain games over others.
Regardless what happens, I'll be the first to say that I'd happily participate in a new format for the dungeon and make some kind of new team. I'll be sure to give input and suggestions as they come to mind, too.