It's not a matter of the converter/charger being new or not. There are two "reverse polarity fuses" on the converter. They "blow immediately" if there is any reverse polarity on the battery cables. If, for instance, the POS cable happened to contact the NEG battery terminal and produce a spark, that "instantaneous spark" is enough to blow the reverse polarity fuses. That action (blown fuse) will disconnect the battery from the converter and stop all charging. When that happens, the battery will drain down and the lights start dimming until they stop working. Then, if you disconnect the load (turn the lights off) the battery will recover slightly, and the next time you turn on a light, it will work and start going dim again until it stops working.
A battery with a damaged cell or a shorted cell (like from freezing while discharged) will act the same way...
From what you're describing, either of those conditions, a bad battery and/or a blown reverse polarity fuse will cause the symptoms you're describing.
As a note, when replacing the battery, it's very easy to blow the reverse polarity fuses while connecting the battery. Now, the battery cables that connect to the converter/charger are the other end of that same cable/same circuit, and it's just as easy to connect the converter/charger (even a new one that you're replacing) and cause the same condition with the cables which will "instantly blow the reverse polarity fuses"....
As a general rule, if, after replacing a battery or replacing a converter/charger and I have any kind of electrical issue with low voltage, dead battery, dimming lights or such, the first thing I do is check those reverse polarity fuses.... Doesn't matter if they are new or not, their function is to open when necessary and while connecting either end of the cables is an ideal time to produce that spark that blows a reverse polarity fuse.....