My son has FPIES to rice, oats, sweet potatoes, green beans, squash, chicken, and eggs. This means he vomits approximately 2 hours after eating anything containing these ingredients, sometimes to the point of dehydration and shock. He was hospitalized for 5 days at 10 months. I don't want to scare you, but it really sounds like FPIES to me, which means you could do more harm than good by trying rice and especially by trying oatmeal again.
Keep us all updated, please! Also, FPIES is rare and still quite misunderstood/sometimes completely unkown in the medical community, so if your doctor hasn't mentioned it don't be surprised. My doctor diagnosed my son with the flu on at least 3 occasions before admitting there was something else going on.
And before you go getting yourself all scared, know that my son is now almost 3 and is a very healthy, normal child so long as we avoid those seven foods. He hasn't vomited since last Halloween (egg incident).
I tried to post some good resources, but since I'm a new member I can't post links yet. Google the FPIES foundation, and also, on facebook look up FPIES and FPIES United Family Fund. Best wishes!