The 6300 models require a battery to be connected to filter the DC for the circuits that have electronics on them.
The batteries going low with the converter on means there is an "open" somewhere between the house battery and the 6300's DC fuse panel lugs where the battery wires go. You need to address that.
The 6300 is single stage at 13.6ish voltage, so it can't do the 14.4-14.8 volts that many battery types are specified to need to be properly charged. You can get by with it if you choose to just replace the battery every couple of years--that might be the least expensive way to go, depending.
It is not difficult to replace the 6300 with a more modern converter if you know how, but might not be worth the cost of paying to get that done professionally.
