If your doing GAPS it does not leave much room for manoeuvre with regard to foods because it is so strict. I've done it myself or at least began it.
If you have an autoimmune illness then GAPS might help but it's a long road with no guarantee that you will hit on the correct combination to eliminate the cause. It's damn difficult to do, beating biofilms and destroying every last organism inside. The biofilm maintains the permeability of the digestive system, because it benefits them, and the resulting autoimmune problems.
I still suggest that you try Pea protein, just a small quantity to see if it works well for you.
The speed and ease of digestion is key to all autoimmune illnesses. Breaking down proteins before they can cause immune system problems.
Milk/casein is very difficult and slow to digest and solid foods are also fairly difficult. Whereas liquid is much easier. Pea protein powder is liquid but I mix it quite thickly. However it's not complete protein, so ideally needs to be mixed with complementary proteins such as rice. I do use a little bit of rice but I need to limit certain amino acids and I might be best to use just Pea protein.
If you search for bulk powder suppliers in your country, you should find many of the body building sites can supply pea protein. It's fairly inexpensive, a lot cheaper than whey.
You could try liquidizing your food and drink it to see if that helps. It should at least a little bit but using easy and quick to digest protein specifically would be better. Hydrolysed protein would be best of all because your immune system should not react to that. I am guessing that you have stomach acid problems which is extremely common among such illnesses.
Using something which alters your immune system function to lessen your immune system from attacking your body would be helpful.
There are difference immune antibodies which can affect the bodies tissues, IgE is an allergic antibody which I am particularly prone to, it's the main antibody to affect my body. High dose B12 can reduce IgE activity by increasing T suppressor cell activity. I use 20mg/day which helps me enormously.
Sacchromyces Boulardii can be quite effective to alter other antibody production in a more favourable way. It might be worth getting some/culturing some and consuming a decent dose. You can culture S.B. fairly easily with a suitable food source. It does work well with milk but seeing as you have problems with that then you would need to find another food source/culture medium. Potato might work okay but would not set like milk/yoghurt so it would be more difficult knowing when to stop it.