An automatic, self resetting circuit breaker, with a bad battery (Shorted cell) will cause exactly that to happen, here is the sequence.
Battery draws max current from converter which lowers voltage to keep from blowing the fuses on it or damaging it internally. Lights dim.
Circuit breaker trips.. Converter sees excessive load go by by so it ups the voltage, Lights brighten.
Circuit breaker resets, and the cycle starts again
MY converter is fairly large, if the batteries are low it can easily pull 10-12 or more amps... IN fact it plus 13.5 amp A/C can take out an older 30 amp breaker (And did a few times) .
YOu need to have your batteries tested.
Procedure: FIRST, find the negative cable which goes off somewhere (NOT a battery) Mark it with black tape or paint and a piece of paper marked NEG.
Find the positive cables that run off to who knows where (NOT to another battery) And mark them with RED (Tape or paint) and a piece of paper (Sticky note) marked POS,
ON both sticky notes write down the number of cables connected.
Take a picture
Then disconnect negative leads, then positive, and take battery(ies) to battery store for testing.
Home was where I park it. but alas the.
2005 Damon Intruder 377 Alas declared a total loss
after a semi "nicked" it. Still have the radios
Kenwood TS-2000, ICOM ID-5100, ID-51A+2, ID-880 REF030C most times