BMS would only act as a circuit breaker, trip if the current is exceeded.
Closed-loop with inverter, it can ask for a particular charge level. Not sure it can ask for a maximum draw.
And when I said buck converter, that only works if source is higher than destination. DC/DC as suggested could be the right solution.
Small inverter and charger is OK. If you get 70% efficiency, good enough in this application. And both pieces are useful for other purposes.
When lithium cells are paralleled, that isn't considered to balance them. Parallel then charge above the upper knee parallels.
If your little battery is low and big battery is full, there will be voltage difference to transfer charge.
If big battery is 70% or 50%, it can't charge small battery except from below the lower knee.