Charging the engine battery

Actually it depends on whether you have an isolator or a separator. In the case of the former - no, in the case of the latter - yes. We have a separator on our 2008 Chevy 190V and have replaced it once because of it being faulty - not charging the coach batteries while driving. Both coach and chassis batteries are now charged when driving and again on shore power. There is a caveat and that has to do with the charged state of the "local" battery - chassis battery when driving and coach battery on shore power. I have used a "clamp" multi-meter to check this out for my own satisfaction.