I saw both those accommodation options that you have mentioned and they were ok, just basic but it depends on what you are after.
Those accommodation places are up the top end of the island near the air strip. Directly opposite there is the 5*Aitutaki Lagoon Resort which has a happy hour every night where you can get half price cocktails($10), a nice up market restaurant day spa etc, there is also a little cafe there called Kuru that makes the best coffee on the island. The other eating places are at the other end of the Island, 10 minutes by moped.
You can kite from the beach you mentioned buy not many do, on normal trade wind days the wind direction would make it cross/offshore. The lagoon is pretty shallow in that area and the wind can be a little fickle as it comes up and over the Palm trees.
IMO a better spot to kite in that area would be the beach in front of the Aitutaki Lagoon Resort,see photo below
To get to the resort you catch this barge type thing that dose the 30mtr crossing(you can see the distance in the below photo.
The problem you may have is that unless you are staying at the resort they may try and charge you heaps to carry you and your gear across, its normally free if you are going over there for dinner or drinks.
I think you would be better off going out to HoneyMoon for a kite.
Another option is to check the web page for Gina's cottages where I stayed as she also has some cottages on one of the islands on the outer edge of the lagoon that you can evidently kite from.
As for taking food on the aircraft, i guess you could but the hassle would put me off. You can buy yoghurt and cheese on the island from any of the little supermarkets and when you factor in the exchange rate its not that expensive