You would use another PIC with a USART/UART to communicate via asynchronous (no clock signal) serial with the RC. If you are going to get this going for next year, start learning now, it is a lot to learn. Most of the PICs with higher pin counts have a USART module, and a few of the smaller ones do too. If you have good resources, you could use one of the PIC30F's which are the DSPic line of chips. Then you open up a whole new can of worms... Quadrature encoder interface... hardware 17bit X 17bit multiplier... multiple work registers... CAN bus.
Frankly it would be quite feasible to take a 30F and make an 8-bit parallel bus to it with the IO's, and a few more control lines. You could then hook up all your sensors to the 30F, and have the RC send it the OI data, let that do everything, and then send back PWM and relay values to the RC, which outputs them to the victors.
Though 6-weeks is one heck of a short time to do that, let alone build a robot to go with it.
Oh and about IFI's hint at 4x more memory, the current user processor is an 18F8520. That can handle 16k instruction words, and 2KB of ram. If IFI goes to the 18F8720, there is up to 64k instruction words, and 3.75KB of ram.
If you are feeling really lucky, they are pin-compatible, and all you would have to do is change the processor definition in the program, and you might be able to desolder a 8520 and replace it with a 8720. that would give you 4x the memory if you could get it to work, however you would lose your warranty, and for some reason I don't think FIRST would like it. So maybe I was wrong earlier, you could get really creative
