They seem to be deliberately obtuse using the term "BSM"...
You are correct the Sterling handles the DC to DC charging from the van electrics only, it's essentially an independent battery charger just powered by 12v not 240v
With a basic lead acid battery you can get away with just powering the leisure electrics on hook up (240v) with a power supply at about 13.8v which is the reference voltage for a running vehicle. The lead acid will charge and self regulate - it's not an optimal charge but it's simple and robust so was the norm for many years. So this is what these all in one systems "charger" is.
With lithium batteries they do not self regulate and have a BMS (battery management system) built in. To charge them effectively you need a dedicated charger that understands the different stages of charge a lithium battery needs to go through.
So folks with these all in one systems will generally remove the fuse (ideally) or just turn off (less ideal) the built in charger and replace it with something like the Victron IP22 or IP64 chargers, there is usually a 240v fridge supply or similar on the system that can be used. They don't need to be particularly high power as they are on all the time you are on hook up, not just a short burst of driving, so something like a 15A is fine.
In the interim it's best to turn off the built in charger at the switch or fuse. You'll probably be fine most of the time with the increased capacity of the lithium battery.