I've got 3 house batteries and 3 engine batteries...the alternator has no problem keeping the house batteries charge even at idle (i know because i monitor the charge level of the house batteries).
i only need to run the generator (going down the road) for the tv, microwave or my air conditioners....its not unusual for us to run the air conditioners while going down the road (we live in texas & the kids & old english sheep dog need the cool air).
when on my mother-in-laws curb, i run a cord to an outlet (hidden behind the bushes) that single 120vac/shore power line is enough for me to run 1 air conditioner (if we need it) but it also runs the converter/charger which keeps the (house) batteries charged.
you are planning to install a converter/charger aren't you ? the converter/charger takes 120vac current & converts it to 12vdc for both utility consumption & to keep the house batteries charged
heres a picture of some simple 2 function LCD panel meters (blue) i installed - to show my incoming 120vac voltage & the Amperage drawl - our coach is 50A (2 legs of 120vac each) so i have one meter to monitor each 120vac leg.
to the far right i have a VERY simple plug in DC voltmeter that displays the output voltage of converter/charger as well as the voltage level of my batteries (when not on shore power).
'04 Haulmark (M42386) 42', 2 slide, 10kw - Pictures
'04 F350 CrewCab Longbed 4x4, 50g aux tank & gear vendors dbl over