You basically have 2 choices:
1: You can make the car run off 100% solar with no battery. This choice is advisable as i think it will be favored more by the D&T markers.
This would require the need for a custom maximizer, with built in RC connection to control the output voltage for speed control. I can build this for you easily

. If you do run this setup, i would advise you use a minimum of 12 watts of solar power, and also use that 4.5v Maxon you got. That would be an excellent power setup with high efficiency.
You wont be able to run a brushless motor off straight solar because the power requirements are too high, and there is too much energy taken in the speed controller/RC gear as well. So the negatives outweigh the benefits. A straight solar/brushless setup i would only recommend if you had a minimum of 30watts of solar energy.
2: You can have an inboard battery/solar system.
With this setup, the battery will act like a fuel tank, and can provide the power required by the BL setup and RC gear during peak loads, and the solar panel will recharge during light loads.
Whichever way you choose, ill be there to help out.