Sorry for the late reply, allergies and intolerance are a complex subject. Carelesstalkcostslives is correct, 'You can only test for IgE allergies ( quick reactions): non-IgE allergies won't show up on blood tests, and can only be diagnosed by excluding and reaction on reintroduction'
I have not personally met a dietician that works on the NHS regarding intolerances (only IgE allergies) so it's interesting that you carlesstalkcostslives have found one that works privately. I recommended a BANT registered nutritional therapist as opposed to just a Nutritionist as BANT registered are regulated by a governing body and have approved qualifications.
I would not recommend cutting out large food groups without some professional help.
With regards to Cow's and goat/ sheep intolerances as on my blog. I have personally found cow's milk is more difficult to tolerate compared with goat and sheep milk. The proteins are larger and more difficult to break down. However, every child is different. The only way to truly know would be to cut all out and introduce one at a time leaving adequate space in between to note any response. Plain live Yoghurt and kefir, due to thier probiotic and fermented nature are generally better tolerated than milk, cheese, cream etc.
Just out of interest - I as a child suffered from reoccurring glue ear - which was eventually linked to a cow's milk intolerance - but did not show on a skin prick test. I'm fine with goat and sheep milk products - though I don't consume large amounts. My daughter can't tolerate gluten - it makes her very grumpy and interrupts her sleep - however she was negative on a coeliac test.
With regards to Rice milk and arsenic. The arsenic comes from pesticides used - I would always chose organic rice milk. In our household we rotate milks - almond, rice, oat, and goat so we get a variety.
I also find experimenting with mixing different milks together to create the right consistency and taste for your child can be useful.
Anyway, sorry for the long post - just a few ideas. I hope you get to the bottom of your DD's sleep Pickledlily.
Best wishesx