The separator will not engage if it senses the charge system is below 12.8 v being drawn down by the truck and camper batteries. This will prevent the camper battery from further drawing down the truck battery. However, with the truck running, the alternator should bring the system voltage above 13v, thus engaging the separator and connecting the alternator to the camper.
It is highly unlikely your camper (lithium) would be that low, even at 70% capacity. Lithiums have a very flat and stable voltage line all the way to 20%.
My guess is, either the separator is defective and not engaging or the wire from the truck is too small. Try running the Lithiums down to 70%, then start the truck up and measure the current going to the Lithium from the truck.