You stated twice that your battery was charged by the inverter. An inverter does NOT charge the battery. A converter charges the battery. There are some high end inverters that have a built in battery charger but they are not very common as standard equipment on most RV's
Since you checked the connections and found them good, I suspect you either have a bad battery or it is not getting fully charged.
Another possibility is you have a converter and a whole house inverter. In this case, the inverter is powering the converter and the converter is charging the batteries that are being drawn down by the inverter. Perpetual motion doesn't work and will quickly draw down a set of batteries. If you indeed have an inverter, turn it off before proceeding to take it out of the equation.
You also keep giving test results but unless I missed it, you have not answered questions asked such as what make and model of your converter? Someone else posted the "12 Volt Side of Life" link. I suggest you read it and part 2 as well as it is very helpful.