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No Differential Required
You really don't need a differential to do what you want. A while back I used to own a Suzuki 4-wheeler. At the time, virtually no 4-wheelers existed with a differential (I don't know about these days). The point is, you can have steering with no differential - it's not that bad.
For cars, the differential makes the steering a bit more stable, but one of the biggest reasons for the differential is for tire wear. Since you're building a short lifespan robot, I wouldn't worry about one side slipping a little bit.
In fact, I would argue that a differential could be a bad thing. Have you ever seen a car with a standard differential on a slippery surface? What happens is that the differential causes more power to go to the wheels that have less traction (that means you'll have one wheel that's spinning like crazy while the one that has tractions is sitting there doing nothing). That's not good for this competition.
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