I don't have a "best" for you, but I'll share some I wouldn't do again. We had the regular dining plan and I have small kids, so we did a lot of the places that have buffet types of things (they're just easier with kids). I'm not a huge eater anyway so I never get my money's worth at a buffet, but it was really not great for GF. They tell you what's GF and what's not, but there are so many things in the buffet that I couldn't remember everything. I had read some good things, but my experience didn't go so well with table service. It made me just not want to do buffets at all, but I'm kind of stuck with it because of the family.
QS experiences in the parks were pretty good. Actually, MUCH easier than I expected.
Epcot-
TS: I can't remember a thing about Akershus but I think it was ok for gluten free as long as you like the menu. You probably don't want to eat there unless you want to see princesses during dinner. It's a madhouse every time. Biergarten was pretty limited for gluten free and I didn't eat much. Our server was really nice but I wouldn't go back. I'm not a big fan of sausage and I left still kind of hungry.
QS: Tangierne Cafe, wouldn't try it again, they were pretty clueless, and my husband actually speaks Arabic and explained to them in Arabic, and they still didn't really get it. My husband loves this place so I got something and had them hold the Pita bread (this was ok for me, but probably not for you). The American Pavilion QS restaurant did a good burger and fries. Sunshine Seasons was kind of overwhelming with all the different options and I think I had a rice crispy treat or something.
AK - Tusker House had a lot of interesting gluten free options in the buffet. I'd go back, but not as a "my only table service" meal. I've eaten at AK Lodge at Boma and it was fantastic but I wasn't yet gluten free when I went there. I like eating food I can't really find at home, so I'd go there, but you pretty much need a car.
DHS - Hollywood and Vine for breakfast, I basically ate some fruit, scrambled eggs and those fried hash brown things they said were gluten free. They had nothing else. I think breakfast is supposed to be the best meal there, so I wouldn't go back. Even when I could eat gluten, I still felt like table service breakfasts were really overpriced. Minus the mickey waffles and it sucks even more. Never in the whole World did I find a gluten free mickey waffle, and I asked.
QS - The pizza place was easy for GF.
MK - I know you said no MK, but the best meal we've had in all of WDW ever (keep in mind we have always gone to the kid friendly restaurants) was at the "Be Our Guest" restaurant. Great service, great food, great atmosphere. It was awesome for gluten free, too.
QS - Columbia Harbour House and Starlight Ray's were both easy for GF.
You didn't say where you were staying, but I stayed at POR and the quick service wasn't great for gluten free. They seemed totally clueless when I asked about it a few times, and I winged it based on internet information. I don't have diagnosed celiac, and I do not have to worry about cross contamination, so I could wing it and get by. I would not plan to eat there at all again. It's not worth getting sick.
We ate at the Grand Floridian buffet place and it was pretty terrible for GF, too. I left hungry. I think I basically had salad and a piece of meat. I guess it was a good "low carb" meal. Had to do it for Cinderella.
You could print out the traveler's cards that explain Celiac disease so that wherever you go, they know you're not just one of those people who are trendily ordering GF but then planning on eating a dessert with a wheat crust.