Without seeing your unit, it's mostly guess work to narrow down the actual problem. Here's my take.
If you're having a charging issue, I would look at your battery, Assuming it's either a liquid lead acid or an AGM battery it could have been overcharged at one time due to a defective charger regulator. Or ....the battery may have been ignored during a winter or two and when the cells weaken their charge the electrolyte tends to go watery. That watery solution can freeze and warps the plates resulting in the plates shorting out. This might give your regulator a false indication of either being open and not taking a charge or closed circuit and showing a full or partial full charge, when in fact.
I might add. Your battery might be fine and it's something else totally different. But your battery is a great place to start and part of the process of elimination
Disco your battery (one terminal) and place your voltmeter leads on the terminals. If your static charge is below 11volts then chances are your battery might be toast. "Static" just means the battery at rest and not hooked up to anything and not under charge.
If it's 12.3 or above chances are it might be ok.
Now, put the terminals back on and turn on the converter and measure that voltage again. Turning on the converter should be as simple as plugging in the trailer to shore power.
A battery that is accepting a charge should be around 13.8 to 14.2 volts sometimes even slightly higher. That's called a "dynamic" charge (actively charging). If there is no difference between static and dynamic chances are you charging system could have issues.
Batteries have a typical life span of 5 or 6 years. However, I have seen much lower and I have also seen batteries last for many more years. Believe it or not my Explorer battery lasted....11 years, almost unheard of.
Maybe your battery needs to be replaced but checking these voltages can certainly help. Any decent automotive or RV shop can do a load test on your battery and yield the same results.
Another test: after the battery is fully charged (according to your inverter) unplug the shore power and turn something on that should be using the 12 volt powering. EG that exhaust fan over your stove. Reason I say that is if your battery has a shortened capacity it won't take too long to actually hear the RMP of that fan slowly drop.
FYI, during the cold winter months I plug my shore power in once a week for a day or two just to keep the battery peeked up.
I hope I helped a wee bit
Jerry & Debbie
Ford Explorer 4.6 Triton V8, AWD, 4x4
1985 Sunline Saturn T-1350
Ham radio VE3JCJ, VHF, UHF and HF