Yes.....Almost all new coaches use the charger to charge both the coach and starting batteries. Somewhere around the time your coach was built, it was a toss up. Some coaches did and some didn't. If you know for sure that your coach was not designed to charge your starting batteries from the charger, then it's easily rectified.
There are three products out there that you can add. The cheapest unit is the
Trik-L-Start.
The one I prefer is a little more expensive and is made by
Xantrex..
The last one is made by
Magnum.
All three EASILY connect to your battery banks. They automatically charge the starting battery when the coach battery is charged. Since your generator powers your current charger, your batteries will charge when the generator is on as long as you add one of the combiner units above.
Many will just add a wire (kind of dangerous) and others will add a separate battery charger. If you want to do it right and not have to worry about it again, add one of the above units. Don't be intimidated by any of the units. They are VERY easy to install, just connect a couple of wires.
Don & Mary
2019 Newmar Dutch Star 4018 - All Electric
2019 Ford Raptor Crew Cab