OK, this probably sounds stupid, but it's just an idea I had. I'm not a drivetrain expert but maybe this isn't such a crazy idea. What if you built a robot that had some wheels connected with tank treads, and others that were not? Perhaps even a tank tread + omniwheel system? It gives you a combination of higher traction and not-so-bad maneuverability. For this game it might be beneficial to have treads on the front and wheels in the back, to navigate the barriers while still retaining maneuverability.
I'm not saying I think it will work, just throwing the idea out there for your consideration. It's not a new concept.
Does anyone think this might work?