Generally speaking, connecting multiple converter/chargers to the same battery bank work OK together. So, leaving the Magnetek in place will probably work fine. However, it is an old design and has many possible failure modes. Replacing it with a new one designed for lithium battery charging would greatly reduce the risk that it would fail or that it would damage other things.
The lithium battery manufacturer may have advice worth listening too. Battle Born has tested many converter/chargers with their batteries.
I looked at schematics for the 6300A chargers. How they are configured varies a lot depending on year of mfg and individual model number. The charging function can be disabled for each one. Probably disconnecting the red wire from the control board will do the trick. However, there are variations and complex circuits that may be affected. If I am wrong, I promise to give you double your money back.
The battery - converter function is switched by a relay. You must maintain that function for the battery to supply house current to the fuse distribution panel when converter is not powered.
I see some models have a filtered and an unfiltered output. The filtered side remains connected to the battery for filtering and the other side supplies "noisy" 12 volt output. Using the lithium batteries for the filtering effect may cause problems due to interactions with the controllers built into the lithium's. The noisy side may not work well with some 12 volt appliances. If it were my rig, I would simply replace it with one recommended by the battery manufacturer.
Disconnecting the red wire may disable filtering and may disable all the filtered 12 volt branch circuits. I was not willing to follow down all the different models schematics.
Good luck and happy trails ahead!