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Per the op's cost of an "on-airport" rental, the cost difference is $5-700.
And re "shuffling," given that the on-airport rental places aren't on-airport and involve a van/bus ride, "shuffling" is inevitable. Particularly given the op's load of people and their stuff, having a cab help shuffle and take door-to-door would be very helpful.
If the car rentals were a walk across the street, it might be different, but I continue to think the best plan might be a) take family and luggage via cab to the Avis off-airport outlet; b) the night before the flight, take family to hotel and check-in, c) take care back to LAX (assuming no drop-charge), d) take Avis van back to airport (or to hotel - I know that Hertz has a shuttle both to the airport and to the near-airport hotels), e) take hotel shuttle back to hotel, f) have cab take family and stuff to airport on day of departure.