You have a lot of stuff going on. My solution with electrical issues is to divide and conquer.
You have at least three components in play. Two batteries and a converter that all provide 12 volt nominal power.
First disconnect one terminal at each battery and plug in the main cord. The converter should provide about 13.6 volts very steady. All systems should work.
Assuming that works connect the two batteries and let them charge 72 hours. Check twice a day for excessive heat or gassing that would indicate a shorted cell. Voltage should be same as the converter at about 13.6 volts right on the battery terminals. If batteries are low on charge the voltage should steadily climb to 13.6 volts over 2 to 6 hours.
Assuming that all seems good disconnect the batteries again and monitor the battery voltage disconnected for several days. Battery voltage should drop from 13.6 volts to 12.65 in about a day and then hold for at least three more days.
Post the results and we can go from there.