I would be tempted to agree with the diagnosis that the two batteries are the problem.
I suspect that the 13 volts that you are seeing at the batteries is from the solar panel.
Best advice is to recondition the batteries again OFF THE VEHICLE but do remember to disconnect the solar panel input to the regulator before disconnecting the batteries. It is possible to damage the solar regulator with the solar panel only connected and no battery as a load. Reconnect the solar panel input to the regulator last.
Having reconditioned the batteries OFF THE VEHICLE leave them at least 24 hours at rest.
It is impossible to read 13 volts at rest after 24 hours.
If the batteries are good you should read 12.7 volts at rest after 24 hours.
Slightly less, but no less than 12.6 (12.5 minimum) volts should be OK.
Do bear in mind that 11.7 volts is considered totally flat for a lead acid battery.
If you need to replace the batteries, do try to purchase ‘maintenance possible’ batteries.
There is no such thing as a maintenance free lead acid battery. If a lead acid battery is charged correctly, it will at some stage in the charging process ‘gas’ which will evaporate some of the fluid. This fluid loss needs to be replenished about twice per year, more often in a hotter climate.
Maintenance possible batteries tend to be cheaper and if looked after and topped up as required will give a much longer service life.
Why not substitue the engine battery for a couple of days. The charger can not harm the engine battery, it is charged by the same charger all the time the green light is on at the control panel.