its not tied into the internals, its wired directly up to the PD output.. the idea being get it as close to the source as possible so its accurate and not suffering from voltage drop.
You must have some fuses in the scheme. And are you using the tow vehicle battery/alternator to help recharge the Trailer battery if you move from place to place on an adventure ?affirmative
also updated my diagram
I think you are saying that you moved the Li Batt + wire from the BP 'in' terminal to the 'out' terminal and that when the Li Bat + was on the BP 'in' terminal the BP did not isolate the charger when the li batt was fully charged. I would not expect that change in wiring to make the BP fail that function.