So you want to basically foam the vehicle, wash it with a mitt and bucket of clean water to keep cleansing the mitt?
I have a foam gun and I always foam the vehicle, then have two buckets, one with clean water and one with soap. The purpose of the foam is to loosen dirt on the vehicle, so when you hit it with your wash mitt, you don't have to work really hard to get all the dirt off. Personally, I would still foam the vehicle and then use two buckets, one with soap and one with water to go back and wash the car. If the vehicle is pretty clean, you probably can wash it pretty quickly. In my opinion, just applying foam and then washing could potentially leave some dirt behind.