Do you have access to a drill press? If you do, drilling holes between the axles slots on the underside of the body to lighten the body is a lot easier. You really want to take as much wood out as you can without making the car to fragile, so start with small drill bits and work your way up. If you drill all the way through and than cover the top with some thin piece of wood, or just tape or sticker (cut out piece of A4 sticker sheet), I would cover the bottom with aluminium duct tape (not the common ducktape!). If you don't have a drill press, free hand drilling might be a bit more difficult but still possible, just clamp the car down and take it slow. You could also drill some holes and use a coping saw to take a big chuck out of the middle of the car, when doing this you will definitely need to cover the top. No matter how you take wood out, I would leave a decent amount of wood in front of the rear axle slot so you can drill the holes for the weights. Before you drill the weight holes, weight the car after you removed the center wood (include the wheels and axles) to see how much weight you need to add . This way you know how many holes you have to drill for the weights. You can either drill the holes from the side or back, you could do it from the bottom too, but then you can only put the weights in front of the rear axle hole. You could also use txchemist idea's, that way you don't have to drill weight holes. Personally I like the denser weights, however you have to be careful that the weight will be too high for you car. What you can also do, like txchemist suggested, is taping cube or even cylinder weights (if that is all you have) together and securing it to the back of the car, however make sure it's secure (and stays secure during the race!) and doesn't change the length of the car. Good luck and have fun.
Maybe for next year, if your scout wants to do a design car again and still wants to have some speed, make a flat thin body and buy some balsa wood, which he can shape in any form he likes and put that on top of the thin body, although still not the most aerodynamic you will have better weight optimization.