I know 22Kauai rented from a local place when they stayed for 3-4 weeks last summer, but I can't remember the name of it. You can message her if she doesn't chime in here. Google tells me there are two options - http://www.islandcars.net/ and http://rentacarkauai.com/
Those cars are definitely nothing to brag about (they are not the shiny late model cars you'll get from the big companies) but the price is right, and she said the people were super friendly and helpful.
I agree with aldergreen - you don't plan to go anywhere besides Princeville and golfing? That's not how I would spend a month on Kauai, but to each their own...
Edited: 26 October 2017, 23:30