no, it shouldn't. a bad battery connected can put a lot of drag on a converter.
if u have a voltmeter, see if u can measure the voltage in the DC fuse panel. check it on both AC and DC. there may be some AC component there but it should be very little. don't know what would be too much, but would think 1 volt AC would be too much. what i'm trying to figure is if the converter may not be offering enough filtering. the converters abt 20 years ago didn't do a very good filtering job. the battery will also act as a filter (get rid of the alternating current pulses). see if u really have 12-15 volts there also.
the frig acting like it just started makes me wonder if there is wrong with the DC. the control board on the frig runs on DC.
while i'm rambling, check the AC volts inside the trailer. that can cause a low converter output voltage.
the converter is usually plugged into an AC outlet (in my vast experience with 2 trailers in my lifetime, they were both plugged into an outlet). check that plug and see if it is in all the way.