My first thought was for Citricos, but then you mentioned the 4 year old. Citricos, IMO, is probably the dullest restaurant on property for a young kid, so I would suggest skipping that one.
I am not sure which restaurants do special NYE dinners and which ones don't (plus that can change from year to year). I would first suggest Flying Fish at the Boardwalk (for us it ties Citricos for best food on the dining plan) and is pretty easy to get to (from AKL, take a bus to DHS and then either walk to the BW, which would take about 20 minutes or take a boat). There should also be some performers out on the Boardwalk after your meal to watch for a while before heading back to AKL.
I know you said no Jiko and no Boma, but are you interested in Sanaa at all? The food is better than some of the Signatures, IMO. The restaurant is nice and is well worth the 1TS credit. We also like the fire pit and look out area better at Kidani than at Jambo. It's also nice because you won't have to worry about large crowds and transportation.