I would call the LG CoC as I am sure they could give you a few suggestions. Many of the cabins on the lake, including the ones we rented, did at one time have fridges which had to have ice delivered and kerosene lamps but most have been modernized as that is what most guests want.
Camping on an island is awesome but it's a hassle getting out there. We used to rent big canvas tents which went on wood platforms. There are water taxis to take you there and back. They will haul a canoe or some other small boat as well. It is roughing it but a real swell experience(
).