Improve HVAC in single upstairs room with vaulted ceiling
I would adjust the dampers so the ones in the smaller rooms are not fully open this will put more flow to the larger room. Things to think about that can reduce flow to a room is the bottom of the door jamb tight to the carpet or floor this can act like a damper once the room is pressurized the flow will be reduced with the damper open. If this is the case trim the bottom door, I state this because the return air duct is usually in a common area for each floor. Your HVAC guy is correct a larger room with less dead air space (attic above) will take longer to cool, BUT a properly size duct(s) and damper control they can be setup to cool at similar rates.