Fish would be fine.
I think crackers are 1.5g of fat per serving? So if the soup is low enough you should be fine with a few crackers.
The reason your doc may be focused on your gallbladder is because it can cause problems with your liver if it is inflamed or a stone is lodged - it's also easier to fix (sort of). Sometimes thyroid tests can be affected by bad infections, so if anything else looked weird on blood tests that may be why the thyroid issues are being put aside for the moment.
I'm surprised you're able to handle grilled cheese. When my gallbladder and GI was(is) acting up, that didn't settle so well for me. Tomato soup was always what I would eat. Generally at that point I had diarrhea so the sodium and potassium in it was good, as well.
Just a little warning. If you do have to have your gallbladder removed, it may not fix all your digestive problems. While it's been less than a couple months for me, the foods I can tolerate are the same ones I ate before the surgery - meaning, fat is still a problem. Though, I can at least eat eggs now.
I still have food triggered diarrhea, GERD, stomach pain, and (this doesn't make much sense) right side pain where my gallbladder was after eating fast food yesterday.
The gagging and dry heaves sounds a bit odd. I don't see too many mention this as a symptom and I certainly never had it (unless very nauseous and vomiting - generally from something else). Have they considered an endoscopy to see if anything is going on in your esophagus? It's not uncommon for gallbladder issues to cause problems with GERD, and GERD can cause those symptoms in some people.