Because you are in a trailer, you will not get much charging from the alternator. (The cables are small and the distance is great). Yes you will get some,but…
If you want good charging from the truck alternator you need to run new wires (probably #8 or#6 from the trucks power to the an Anderson power connector (for a good disconnect point) then to a Dc-Dc charger. That way it will charge correctly and at the rated power. If you are thinking about that -look at the Victron isolated Dc-Dc charger 30a (you need an isolated charger).
The main issue with charging lithium’s from the alternator is it never stops when the battery gets full. Charging a full battery at 14.?v for hours while you are driving is not good (and potentially very bad) for the battery.
In MotorHome’s where we have thick cables that can handle 200a, and a 200a alternator and a big lithium battery- the alternator gets ran at 100% all the time and can overheat and burn up.
If the bms kicks off when the alternator is running, you need a place for all the electrons to go, most units are also charging a lead starter battery and so the electrons go into the starter battery, if your alternator does not have another battery- that’s when the diodes get fried.
200w of solar is not a lot- I would look at expanding the solar first (better bang for the buck).
Also, I would install a switch somewhere so you can kill the power from the alternator charging the trailer. So if your battery is full you can stop it, but that trickle from the alternator is nice when needed. You probably need to measure the voltage and amps that the vehicle can charge the trailer.
One REALLY nice (in my opinion needed) piece is a shunt based battery monitor- like a Victron Smartshunt or BMV712. This way you know how much power you really have in the battery. It is also good for figuring out how much power different stuff takes to run.
Good Luck