First, a new battery does not automatically preclude a bad battery. Second, a bad battery can still meter voltage fine, but when placed under load, the voltage will drop. The "full charge" indicator on these types of chargers basically tell you when it cannot squeeze any more charge into it. It should not be taken to mean "all 15 Ah" are in the battery.
So, the problem could be the battery, the charger, or the wiring harness. The battery can be tested under load at most auto parts stores. The charger should meter, at the battery end, 13.8VDC or higher when plugged into the wall.
The wiring harness will be a little more difficult to trouble shoot. Make sure all connections are tight, and there are no cuts in the wiring. Double check the fuses to ensure they are fully seated. For the purposes of trouble shooting, the charger can be plugged in to any set of Anderson connectors, as can the inverter. If you get different results, then there may be a bad section of wiring on the lesser performing lead.
As always, we will be glad to take a look at the system and let you know what the repair needs are.
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