I'm am an extremely picky adult and I never really had a problem with the counter service restaurants. A lot of restaurants while I was there let me order off of the kids menu... I actually preferred this because they had smaller portions and healthier options. Hungry Bear in critter country was a favorite stop of mine. I'd get the nuggets and they'd come with fries and apple slices... And then I always brought water bottles, carrot sticks and protein bars. I didn't ever feel tired from eating greasy food because I had a nice balance, and I was able to cater to my unfortunate case of picky eater.
You said you're really not looking for table service restaurants like Ariel's grotto, so depending on how picky they are there are a lot of different options. If you stop at a restaurant they could always run to a different one get their meal and bring it back to where the main group is. By the time the whole group has ordered and sat down they should be back to join you.
Village Haus was really the only counter service I didn't like the menu... Even the kids menu didn't really offer much for me. It was pretty much burger, pizza or turkey samdwhich... And I'm not big on turkey and I don't eat beef at all. So all that left me was pizza.
Also like others said above some of the snack options can be meals, the pretzel with cheese dip was always a good lunch option for me. It was big enough to last as a lunch ( again I always brought water, carrot sticks/fruit, and granola or protein bars)...
If your group does want to eat at a nicer restaurant like blue bayou just ask the picky eater if they'd be fine sitting it out... As long as they're old enough to be alone and wouldn't mind it they can always run off with $20 and their cellphone and meet you after. They'd be able to eat something they'd like and maybe squeeze in a ride or two. I have a big family and they understand and know how to work with me... so that'd be something we'd do, I'd go off and do my own thing and they could have their meal and then we'd meet up after. I would t feel excluded because we literally will spend the whole trip together anyways, but just make sure your picky eater is as independent and free spirited as I am, make sure they won't feel like the black sheep being away from the group.