I fixed a similar problem I was getting 13.6 from the converter but not to the batteries
After checking the relay and fuse (1 and 4) I checked the solenoid (2) seemed to be the problem and I just connected the wires to another one to find out. Removed the old one and securing the new one
Two other times it was a fuse in the panel Not one in the long row of fuses, one of the two fuses beside the row.