Unfortunately we won’t have access to a kitchen area. My husband is going for a conference and his boss is letting us tag along; most likely staying at either Swan or Dolphin. And if not one of those, it will be another on-site hotel (we’re still waiting to hear where they’re putting us). We won’t be able to upgrade to anything more than a standard 2 bed room either. So I’m really trying to see if eating out is going to be a possibility.
OK, then. I'll see what advice I can offer based on my own experiences, although my diet has never been quite as restrictive as yours is, but I'm a vegan, so I know how to be careful at WDW.
The very creative chefs at most TS restaurants at WDW will come out and talk with you to see what they can come up with. They're extremely careful about allergens, so when you speak to the chef (I think this person is called an executive chef), let them know that your chicken or turkey has to be cooked separately from anything else, for example.
I've had TS restaurants make me steamed veggies, even places that seem like they wouldn't be able to do that. And I think steamed veggies would work for you, although that's hardly a full meal since you can't have some kind of grain with it. But at least it's a start.
Can you have oatmeal? It's gluten-free but I'm not sure it fits your no-refined-grains requirement. If so, most places do have oatmeal, it's made without anything but water and some salt (maybe not even salt) and you could put some banana on it. If you could pick up oat milk at a grocery stop, that could go with your oatmeal, assuming you can have oatmeal. Assuming the oat milk would fit your other dietary requirements.
Even without a kitchen, there are coolers or mini fridges in the rooms and most fruit will keep without being refrigerated. I highly suggest getting groceries. I've had trips where I shipped myself certain nonperishable food items when I was on a particularly restrictive eating plan and you might consider doing this as well, assuming there are nonperishable food items that fit the bill for you.
From reading your post, I'm thinking that at home you most likely have something like grilled or baked chicken and steamed veggies for dinner. I think these are things that could be rustled up for you at almost any TS at WDW, although I wouldn't count on the World Showcase restaurants at Epcot to be able to do this for a variety of reasons, mostly because they're preparing country-specific food items so they might not have a variety of ingredients on hand.
You are going to have to plan. I'm sure you don't want to be caught wandering around looking for someplace to eat and not finding anything, and there's very little you can snack on (no popcorn, no pretzels), although all the parks have places where you can buy fresh fruit. But if you're nowhere near these places, you'll be doomed to walk another 3 miles to get there. Plan ahead.
You can bring food into the parks with you. Maybe put a couple of apples, for example, in your bag, so you have emergency, I'm-famished-and-I'll-scream-if-I-don't-eat-soon food handy.
I don't know if this has been a help. It's definitely putting a damper on your WDW trip but it can be dealt with. You just have to make certain concessions, definitely have a grocery stop, and plan ahead.