I grew up in Chattanooga but haven't lived there in about 7 years. Coolidge park
used to be a really nice park but I have heard its gone slightly downhill in the past few years so I can't promise its still in its prime.
If you were wanting to rent a cabin I grew up next to Cloudland Canyon State Park
, which is where we always had our family reunions when I was a child. Its fairly close to Ruby Falls and Rock City which are some of the more popular tourist destinations of Lookout Mtn. But, Cloudland Canyon is about 30 minutes from Chattanooga which might pose a problem if you guys wanted to go to a restaurant or somewhere decent for dinner.
I would look into those at least to see if they would suit your needs. I can't think of anything else off the top of my head I'll let you know if I do.