Nut butters on bread/crackers, PBJ. Cheddar cheese in block form lasts a long time unrefrigerated, cured meats like pepperoni. Carrots, cucumbers, tomatoes all keep pretty well unrefrigerated.

Oatmeal

Tahini (you can dip your pita in there)

Beans, canned or dried if you’re able to cook. Lentils also.

Chia seeds - some bags say to refrigerate them but as long as you’re not letting them sit there for a long time you’ll be fine. Same goes for tahini actually.

Nuts and seeds

Could also get other breads or maybe corn tortillas or something

Edit: I suppose by no heating element you mean you can’t cook. So with the oats you’d probably have to get old fashioned oats and do overnight oats or something just using water. Canned beans don’t need to be cooked so you can use those.

Edit 2: ugh I actually don’t know if the overnight oats would work. Everyone I know who makes them uses a fridge. You can eat old fashioned oats “raw” though, since they’re not actually raw but they’ve been steamed. A little weird though. Though you could make a cereal with them or something, like muesli. Actually they sell packaged muesli in the store, that might be a good thing to try, and you could get oats that way. I love getting a box of Alpen.

Other than what you’ve mentioned. When I go hiking for long periods of time away from facilities I often go to these staples:

Breakfast: premix portions of dried milk powder and oats to use as instant porridge, add in some salt some sugar and some freeze dried raspberries for taste.

Lunch: a hard tack / cake thing made with flour, oats and suet flavoured with dried fruit. Keep forever and very energy dense and quite tasty.

Dinners: (other than dehydrated meals which are always good for expeditions) couscous. Bulgar wheat. Rice, pasta, noodles. Dried meats and fish. If you’ve got a dehydrator you can make things like dehydrated roast peppers or other veggies. Things like butternut squash, potatoes, onions etc don’t actually need refrigeration, just to be kept cool and dark. Also if you’re not travelling anything you can grow - herbs like parsley, chives etc add so much to a meal.

Snacks: flapjack, nuts, chocolate, dried fruit (dried mango is my all time favourite) fresh fruits etc.

I'll sound like a broken record on this site, but rice and beans can be made in small enough portions that you usually don't need to consider FATTOM.

Is buying a rice cooker a possibility?

If that is doable then it opens up a whole world of possibilities for cooking - with the button pressed down you are able to fry small quantities of food and with the button up it's basically a slow cooker.

Bean salad with nachos