I use to have one of the edgestars fp430, at 40 degrees the fridge will use about 25 amps in 24 hours. I measured it with a dc wattmeter, even when I thought the fridge was cycling too much, it always use about the same amps. So your system is more then adequate to keep up with.
I think the battery is done for, it looks like it got discharged down low to many times and not fully charge backed up. If the battery ever goes below 11 volts, it usually wont last long. It will behave like your battery, charges quickly to 14.4 volts but then quickly discharges once you start putting loads on it.
When I bought a lead acid battery the person at the battery store told me that with solar I would be trickle charging my battery most of the day, he recommended I charge it at high amps every once in a while so it would last longer. One thing I would definitely check, is the voltage difference between the battery terminals and your controller. If you have voltage drop, your controller can read 14.4 volts but your battery terminals might read 13.9 volts. I encounter that so many times I always compensate by raising the bulk setting to 15 volts to get the battery terminals to 14.4 volts. If you had voltage drop, your battery was never getting a full charge and that will lead to an early demise for the battery capacity.
With solar always keep your float voltage high, if you can get it to 14.4 volts do it. Some controllers shift to float too early after reaching 14.4 volts. The sun goes down before solar can fully charge a lead acid. 14.4 volts all day long won't harm a lead acid, it will actually make it last longer.
If you decide to get an mppt controller, I recommend the ecoworthy 20 amp mppt. (cost 100 dollars), mine has been running non stop 24/7 for almost 6 years. The reason I recommend is because you can adjust the bulk/float voltage on the unit itself to compensate for voltage drop. And it has a built in lcd screen.