My almost 2 year old is also the exact same way. We started noticing ocasional things while I was still breastfeeding. It seemed that after certain things I ate, he was really fussy, etc. After he weaned around 16 mos, and we tried to do whole milk, the diaper rash and fussiness were horrendous. It took me a couple weeks to figure it out, but being a nurse, I came to the conclusion that I thought it was dairy. We stopped dairy, and went to almond milk, and after about 5 days of treating the diaper rash, he finally cleared up, and got better. We continued to try and re-introduce milk about every 2 months, and each time the same symptoms reappeared, and we would go back to almond milk. I finally started on lactose-free milk for him about 6 weeks ago, and he is doing awesome. So, he gets lactose free whole milk now, and no diaper rash. He seems to not have much problem with yogurt and cheeses (which is common), but if he eats too much of any of it, his stools will get very runny, and the diaper rash comes back quick.
We figure he may grow out of it eventually, but until then, we will just accommodate his intolerance as best we can.
The best thing you can do is continue to breastfeed. I'd have given anything to have him still on the breast thru this, that way I would have known he was getting what he needed from me, and we could rely less on all other foods.