DS used to get REALLY wet nappies after drinking it but no red bottoms (he gets red bottoms with numerous foods - carrots, soya, apples, being the main ones - it leaves a burn mark where the poo touched his skin, but he isn't sick or anything with it, but has rarely been sick, even as a baby)
.... try oat milk instead and see how you get on (Oatly, with red writing, in the UK has added calcium, etc). We were advised by dietician not to use rice milk due to levels of inorganic arsenic content!!! I think this is until at least the age of 5.
DS is on a dairy free trial through the dietician and she prescribed (well, got the doctor to) Neocate which is a dairy-free formula. He is on the active one as that is for children over 1.
Good luck with the trial - it's tough going figuring out what upsets their little tummies.