I've been working an idea that is somewhat similar. Basically using four load cells and/or strain gauges (more or less the same thing) to measure the balance at each corner.

The main concept is that you don't really need to measure an accurate weight at each corner, you just need to know what they are relative to each other. For example when the car is balanced each load cell will produce 7mV. This doesn't tell you anything about the weight, but that doesn't matter. You don't need to worry about using an algorithm to convert the signal to a force (weight) since you just want to know the signals are equal.

I have been in the lab playing around with a few designs, but the load cells available there aren't optimal. I've been looking into getting some better load cells, and still playing around with the best way to display the load cell voltages.