Hi Russ:
The answer to your question is no, only the started engine with the alternator working will charge your chassis batteries. However, consider this work-around.
I bought a Battery Minder. It is totally automatic, and trickle charges the battery, does automatic desulfination, and got rid of an innate problem of long term camping, or long term stagnation (not starting the engine). The unit is about the same size as an a/c adapter for your cell phone.
Drill a 3/4" hole from your shore-power compartment to your battery compartment. Take care not to bore into any of the battery cables. The Battery Minder, or whatever unit you choose, has a line with eye connectors that easily slide over the proper battery post. Run this through the hole you bored, I used a rubber grommet to protect the line from the rough metal edges. Plug the unit into the outlet found above your powered line reel. This outlet is where the engine heater is plugged in, which is why you have a plug sitting there.
By the way, do not plug in the cord and leave it plugged in (for the heater). The heater uses a lot of amps, and will burn out if left continuously plugged in.
I put velcro on the Battery Minder and the outlet, to keep the Minder in place.
No more dead chassis batteries, no more worrying. Took about a hour to do everything.
http://www.pacificbattery.com/batteryminder.html
Safe Travels
Rick