A flat floor would be good for drag reduction. You could make it generate some downforce too - probably enough to counter the lift forces generated by the upper body shape. You would need to be careful not to introduce an aero imbalance though; it would be quite easy to end up reducing overall lift at one end of the car and not the other. This would result in some interesting handling traits.
Large amounts of downforce isn't a great idea on a road car for three reasons:
1. When downforce is interupted the grip drops instantly which means you're likely to crash if you're cornering on grip provided by downforce and then the downforce is lost for any reason.
2. Large amounts of downforce adds drag although floor derived downforce is generally less draggy than that derived from wings/spoilers. Large amounts of downforce also require stiff suspension to prevent bottoming out at speed etc. Not great on the public road really.
3. A fully enclosed floor means that you need to introduce cooling ducts elsewhere. Don't forget that on a road car, the air that goes in through the radiator generally exits in to the space below the car. This isn't a deal breaker but it needs a lot of work to prevent overheating (even on a MR2 although that would be easier because of the engine position). Also, a fully enclosed floor might reduce suspension / steering articulation.
Designing a flat floor that generated downforce would be quite involved. You could just slap on some alloy panels to give a smooth surface and that would cut drag for sure. Would it generate a whole heap of other issues? Almost certainly. Worth it? Only you know that...
I would think that you'd need to put a flat floor (without any diffuser etc) on and see how it went. You could then add a diffuser (keep it reasonably shallow though - if you put on a steep diffuser it will likely experience flow separation and be no use anyay). You'll probably need a front splitter and/or dive planes too to try to balance the lift reduction / downforce figures. You might look at the Porsche 911 - I think it has a sculpted underside in front of the front wheels to help generate some downforce. Here is a picture of the nose section of the flying 1999 Merc CLR from LeMans. You can see how a diffuser has been created to try to generate front end downforce to balance the car:
It will require a lot of trial and error but should be quite fun and fulfilling to try out.
If you are more fortunate than others, build a larger table not a taller fence.