i didnt stay at the Cheyenne so cant comment of whether there is a kettle there but there was in our room and thomas Cook's explorers hotel.
There are no sup[ermarkets on the park, theres a petrol station nearby and also a shopping centre one stop on on the Eurostar which may have. I would also expect the crocket ranch to have a supermarket nearby but thats not within walking distance of park
We went in september. We took lots of snack, crisps, biscuits etc plus coffee and cupasoups (if needs be why not take a travel kettle)
At breakfast we made up sandwiches etc to take in our bags and had picnics in the gardens at front of main gate. There were also plenty of shops selling snacks inside the park.
I dont think the prices were as high as you say, but you could also look at maybe getting the halfboard options and booking restaurants for each day. The all you can eat buffets are great and can really fill you up.
we did have bags searched at gates, but at no point did anyone remove food that we had - and all the kids had lunchboxes and flasks which were not queried at all