There is a difference between a full-blown allergy and food sensitivities. Food sensitivities might not even show on a blood or stool test, but irritate your gut enough to cause inflammation, especially when you are still overweight and possibly have leaky gut trouble. Personally, I think it is best to listen to your body there. P3 is actually easy... at least it was for me, since I transitioned into paleo lifestyle.
For now, stick to meat (preferrably grass-fed/ pastured), any fish, any non-starchy veggies, and good fat - olive oil and coconut oil, for example, preferrably organic/ cold-pressed. That gives you loads of options, and you can get ideas looking for paleo recipes.
If you are more than 2lbs over your LDW, please do a steak day first. Then stick to the paleo list for a week or so... definitely until your weight and irritation subsides. Once you are sure that inflammation is under control again, start introducing the common sensitivities one by one. Eggs, dairy, nuts, cheese. Keep a couple of days between that and see what your body does.
For me, this has meant that while I can do some whole milk in my coffee or sour cream on my fish, straight full-fat yogurt or Italian mozzarella are out. The same for eggs... no problem with some pieces in my salad, stomach cramps with a regular omelette. On the bright side - loads of energy, no depressive episode ever since starting, major pain in my ankles (due to an accident) that had me on crutches at times just disappeared... Less inflammation is really a godsend.