The AMT will only charge the van battery under certain conditions. It all depends on what mode it’s set to. Basically, it won’t start charging the starter battery until the LB is just about fully charged AND if the starter battery is in a certain voltage range. IF the starter battery is below around 11V I.e. flat, the AMT won’t attempt to charge it. IF the starter is circa 12.3V I.e charged, the AMT will shut off.
Also, the output from the AMT is very low, so won’t keep up with any high parasitic drain from the starter.
You need to watch the LED on the AMT to see what’s going on.
Have you got the Sterling set up correctly to shut down when the engine is not running?
I have my AMT set to Mode7, lithium/stop-start/high current.