One of the most likely causes of a low battery reading when you are plugged in and your converter is working properly is a bad connection from the converter to the battery. It could be either the positive or the negative at the battery end or the converter end.
Verify your converter output is over 13 volts first by putting your meter leads right on the converter output terminals
, then connect a wire to the converter negative terminal and use your jumper cable on one lead to the battery negative. Does that make the battery voltage match the converter? If so, find the loose negative wire. If not, do the same with the positive lead.
In my case, the negative feed on the converter terminal strip was loose, the converter was putting out full voltage, and the loose connection acted like a load, the battery was down below 12 volts before I checked it out with my meter and fixed it.