You can go with the batteries undercharged for a few days at a time and then do a full recharge. Your problem, as you said, is not enough solar to get it done in the one sunny day. BUT if you have 12v batts you can probably do up one battery at a time with that much solar all day and run the rig on the other battery.
You just might be able to get that one battery full in one daylight period (until later in the year of course.) Do one batt and then wait for the next good day and do the other. Bank them again for overnight or on part cloudy days.
The idea to get full at least some of the time is to desulphate since the last time. So you can do one at a time at different times, as long as each one gets done up often enough.
IMO you don't need another solar controller. Your real problem seems to be not enough of a battery charger to run from the generator. Lots of options there. if you had one, you could do gen /charger in the morning for Bulk and follow that by solar for the rest of the day to get close to full. Also you want a decent charger for the coming "dark" days till next Spring.