Hope you have a good time, chaos! I'm guessing the cabana is a tent-like thing within the park? What a shame you can't take a small cooler in. Maybe you can ask at the gate, and explain your situation. For me, I would be concerned that all the cooling benefit that happens in the car would dissipate by the time I caught up with my group.

Your plan of frozen water bottles is excellent! Suggestion: put a damp towel in the cooler for an all-over cooling effect. Remember to reapply sunscreen! Wear a big straw hat. Straw, not a solid fabric one. Bodies lose the most heat off the head; you need to be able to let the heat go, not hold it in.

If there are scooters to rent, do so, as Doydie suggested! Call ahead; you might have to reserve one.