Your batteries should do OK....just to add to the conversation, I have a residential Refer, a portable ice maker, a mini fridge, a wine cooler etc etc that require 120 volt.... I usually run these all off my 2500 watt inverter but what I have noticed is at idle, I'm not getting enough amps out of the alternator.
Generally I wouldn't worry about this but with recent issues with my Allison Tran's not getting enough voltage, I'm in the market for a higher capacity alternator. The MH originally had a 220 amp alternator(Leece Nevell)...after many issues I ended up with a 125/150 amp that has done the job prior to all the junk I have running. Good thing I run the generator alot!!! :-)
OH, getting ready to add a Norcold 3.2 CF Freezer only(great deal at Adventure RV@ $205.00)...another reason for larger capacity alternator...also have 6 6 volt batteries.