Sure it would work. If your goal is to avoid installing/using a separate generator, however, it is a bad idea unless you plan to drive a couple hours every day.
Running the chassis engine JUST to charge the house batteries is the bad idea. You will find it very wasteful of fuel, and it will be hard on the engine, as they simply are not designed to sit and idle a lot. They are designed to drag the vehicle down the road, and they do that well. But sit and idle, not so much.
Think about this in the sense of firing up a nuclear aircraft carrier for the purpose of pulling stumps. Long enough rope, and it'll work, but there are much better ways of pulling stumps.
