Noel,
You have posed an unanswerable question. This is why you have gotten an assortment of contradictory answers. Without knowing a lot more about the coach and the chassis, there is no way to give you a solid answer as to which will be faster.
First, there is nothing good that happens to an engine at idle. Even if the engine is equipped with a 100 amp alternator, it will not produce that much at idle and the engine itself may be burning off 20 amperes to run the engine management and even more if the cooling fan is electric.
Second, without knowing the manufacturer and model number of the converter currently in the coach, how that will compare to the main engine alternator cannot be established. My tiny coach has a 60 amp three stage intelligent converter. It also has a 100 amp alternator (now) and there is now way that the main engine can rebuild the charge in the house bank as fast or as well as the converter whether it is powered from shore power or the APU.
I can pretty safely guaranty that the APU (genset) will burn less fuel than the main engine will in the process.
Me, I'm a ship's engineer that came ashore several decades ago to work in product development in Detroit. If you care to listen to me, that may serve you well.