When our son was around that age, we got a lightweight car seat and actually installed it on the airplane. Such a life saver! He was used to sitting in a harnessed seat in the car so sitting on the plane in one was no big deal for him. I highly recommend this. We would just tether it to a wheeled carry-on to transport around the airport. Then we used it in our rental car.
This was taken when he was 5, but it's the same seat he used at age 3. We never felt comfortable using a seat from a car service since we didn't know if it was expired or had been properly taken care of. And we never wanted to check a car seat since they throw them around so much at baggage.
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Here's some good info about flying with little kids -
https://csftl.org/leaving-on-a-jet-plane-the-csftl-guide-to-safe-air-travel-with-children/
As for getting to and from the hotel, something like Uber will likely always be the cheapest. But last time when we didn't rent a car, we used Lansky car service. It costs more, obviously, but it was super easy and convenient and had plenty of space for all of our luggage. (Our son is 9 now and we only needed to bring a little backless booster for the car this time.)