I have coeliacs disease and have really struggled with anaemia in the past, particularly as a teenager. What really helps me is to focus on iron-rich protein-rich meals, being very strict about avoiding gluten contamination, and taking a daily multi (which includes iron, vitamin C, and all the B vitamins, all of which are helpful for anaemia).
Examples of iron-rich, protein-rich foods: for breakfast I like tomato and spinach omelette, bacon egg and beans, or make-ahead egg breakfast “muffins”. It’s great to focus on having plenty of red meat for dinners, and doesn’t have to be as expensive as regularly eating steak - I like cottage/shepherds pie, beef stew, chilli con carne, Chinese beef stir fry, lamb meatball tagine, bolognaise, etc. Focusing on mince-based meals keeps the cost down. Just make sure she’s getting a good sized amount of the meat.
Right now your daughters diet as described is quite low in protein and very low in iron, so I’d really try to prioritise this. Bear in mind that plant sources of iron are far less absorbable than meat sources, especially for damaged guts.
If she does get diagnosed with celiac’s disease, you might want to consider using a doctors note to make the school allow her to take her own lunch. It doesn’t sound as if the school kitchen is particularly coeliac aware if they are only serving plain rice to the students who can’t eat gluten, and frankly I wouldn’t trust them not to contaminate the supposedly ‘gluten free’ items. (It’s easily done in a busy kitchen!) Every time a celiac consumes gluten, even small amounts from contamination, damage is done to the digestive tract. Repeated exposure to gluten contamination increases risk of various digestive system cancers, so this is really important to keep in mind.
I know other posters have mentioned oats. Roughly one in three coeliacs are sensitive to oats. This is because even gluten free oats contain the protein avenin, which is structurally very similar to gluten and can trigger an immune response for many coeliacs. I’d personally get rid of the oats, at least until her digestive system is fully healed. Then you can try to re-introduce certified gluten-free oats and see if she reacts to them. It’s important that they are certified, because regular oats contain gluten.
Probably the most important thing to bear in mind is that the damage gluten causes to coeliacs intestines can take months to fully heal. As such, if she does indeed have coeliacs disease, it could be a while until she is fully absorbing nutrients from her food, no matter what you do.