First I would contact the hotel about whether or not they have a shuttle service. If not I suggest hiring a car and driver. With four people and luggage this might cost a bit more than the cab but if you get one where the driver stands at the exit from passport control and holds up a card bearing your last name it might expedite an orderly departure from LAX.
There are cabs that will take all of your family and luggage but you will probably have a higher level of comfort it you do the driver and car routine. I don't know which service to recommend at LAX. Perhaps the hotel can recommend one to you.