Lazy Flamingo 1 sort of qualifies as a fish shack, but only the one in Cap San Village right at Blind Pass. I think of it as an oyster place. They do grouper well and have wonderful oysters.
Have you been to Sanibel before? It is a very laid back but upscale resort and winter snowbird retirement community. Any fish shack here would be highly produced and not real. There are not any "fancy" restaurants on Sanibel. Casual but sophisticated is what most shoot for in the high quality places.
If you have a boat, it might be fun to go to Matlacha, where fish shacks used to abound.
You may like Doc Ford's. It promotes itself as a Rum Bar but they do fish very well, it is casual. The owners is a famous Sanibel author Randy Wayne White. If you haven't read him, the books are highly entertaining and full of Sanibel atmosphere and local lore.
Over Easy Cafe has the best breakfasts by far.
The Bean is coffee central. So Sanibel that it has it's own outpost at the airport. Great coffee and muffins, etc. It's on Periwinkle.
My favorite better restaurants are: Sweet Melissa's and Il Tesoro. They can compete with big city restaurants, indeed, Il Tesoro has an upper east side NYC location. It is tiny so you must reserve. It's on Open Table.
IMO, the restaurants on the island are much improved over the last few years.
There are many others and I'm sure someone will offer their selections.