Glad to help! I will be interested to hear about your Meet and Greet experiences; we've been tempted but haven't used that service yet.
A service that we have enjoyed is the free shuttle from the airport to the George Town Yacht Club at the Barcadere Marina that is located only a few minutes from the airport. We used this option twice during our recent vacation to Grand Cayman and Little Cayman.
We shuttled to the GTYC after checking in before our flight to Little Cayman and also during our trip home after returning from Little Cayman and checking in for our flight to Chicago. It's a much nicer experience than hanging around the airport, especially during the construction. Good food, beautiful ambience, and friendly service. The GTYC has outdoor seating so I think that would okay for your dog, but you should probably confirm that in advance. They primarily cater to people visiting Cayman by boat - so I would think that some of them would travel with dogs.
The restaurant at the GTYC is owned and operated by the same people that own Macabuca and the Cracked Conch and I think that the conch is even better at GTYC - especially with their special tartar sauce.
They open early on Saturday and Sunday and serve breakfast until 11 am for early travelers. We got there around 10:15 am Saturday during our trip home, but we had already eaten an early breakfast at the Little Cayman Beach Resort and we wouldn't eat again until after arriving in Chicago; so they graciously agreed to make their famous cracked conch for us - the appetizer portion is the perfect size and it's delicious! See this link:
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Have a fun trip, where are you staying and diving?