Your inverter is red lighting because the voltage is too low for it to operate. 11.5 VDC is a typical low voltage threshold for small inverters. The reason it operates when you have the shore power connected is because the converter is supplying the 12 -> 13 V necessary for correct operation.
I would guess that your batteries may be shot. The warranty date tag on the battery shows 2006 as year of purchase. That puts them past the typical age of most usable trailer batteries. As was mentioned, charge them fully and bring them to an auto parts house for free testing.
I'd like to make a couple of suggestions: When replacing the batteries, choose two 6V GC2 batteries. They have the same footprint as a group 24 battery. The 6 V's are more durable and tolerant of usual RV use and will triple the available capacity. Two 6 V's are able to operate a 1000W inverter.
If you don't have a disconnect switch installed that can interrupt power from the batteries to the trailer, install one for storage times. This is one thing that TT Mfg's don't install and should be in every RV, IMO. If you don't have one and leave the batteries connected, the parasitic draws from alarms, circuit boards, and stereo units will drain the batteries in a week or so.
Make sure you have a fuse or circuit breaker between the inverter and it's battery connection. Use large cable to connect it to the battery and locate it as close as possible to the batteries.
Hoping you get it all ironed out. Inverting battery power allows for more variability of how an owner can camp in the RV and also allows for greater comfort.
Ask about solar power in the "Tech Issues" section of the Forum. Solar is to battery charging like peanut butter is to jelly.:B