Need some guidance

[JimT]

So you're saying that, while driving, the roof AC through the inverter is dropping the battery voltage reading down to 10.9 volts ?

First thing to check is the alternator itself. If you've been running that kind of load constantly while driving then it may have strained the alternator. Make sure it's still outputting the correct power.

Second thing I'd do is check each battery and make sure you don't have one with a bad cell.

Then check all the connections and grounds.

Finally, see if there's any place you can shorten the length of the battery cables or upgrade to a bigger sized cable. 4/0 all the way from the alternator to the batteries to the inverter would be the best. The longer the run and the thinner the cable the less current it can carry. My Xantrex 2000 watt inverter recommended minimum of 2/0 cables at a length of 5 feet or less from the batteries.

If none of that helps then you probably just need a bigger alternator to supply more power for your application.

Under heavy load the voltage will fall. By itself it's not a problem. The problem is, however, if you operate for long portions of the day like this then the batteries will not be charged by the system. They need to see between 13.6 and 14.3 volts (depending on the type of battery) in order to charge properly. If your loads are constantly so high that they don't spend enough time charging, then you will shorten their lifespan and find yourself with weak/dead batteries.

If you're looking for some high quality cables made to whatever sizes you need, then try https://www.batterycablesusa.com/. I've replaced nearly all my cabling with stuff from them, some 2/0 but mostly 4/0. All much better quality then anything most shops or dealers carry and they are fast to ship.