Keep in mind that your converter charges the battery at 3 amps - not the full output of the converter.
The standard failure mode of that converter is for the fan to die which allows a current limiting resistor and bridge rectifier to fail. When that happens the failed diodes cause a continuous drain on the battery.
You've caught it right before the catastrophic failure.
It's time to replace it with a good three stage converter like a Progressive Dynamics Inc (PDI).
I suggest you go to Bestconvertercom and check them out. Randy is very helpful and will steer you in the right direction. There's even directions on their site of how to swap out the converter.
