My first thought was bad connections or bad ground at battery or convertor.
Does the tops of the battery cells have caps on them that you can take off and check the water level? May needs to fill with water and/or check for dead cells. Pull the battery out and take it to a Auto parts store and get it 'Load tested'. the '13.1v' is showing the charge rate from the converter. when you take the cables off and it drops to '12.6' it the 'hold' rate at that time. Put any type of load (lights) on the battery and it could drop down to 10v.... Your converter will then 'boost' up and start charging at '14.3v'.... This is not good on your convertor if it is 'charging' all of the time (higher than a trickle charge) into a battery that will not keep a charge.
With the cables off the battery, test the cable out put from the converter.... With the battery disconnected, turn a couple of lights on and take a volt meter a see the voltage output from the positive cable.
2013 Forest River Salem Hemisphere SBT312QBUD
2001 CrewCab F-250 7.3 PowerStroke Diesel
SuperChip, BTS transmission, 6.0 Trans Cooler