I highly recommend seeing an allergist - peanut allergies can be deadly! I know several children who are allergic to peanuts and CANNOT have peanut oil in an ordinary setting (like those used at a restaraunt) b/c half the time they don't know WHAT kind of oil they are using. Never walk into a restaraunt and ask them what is in their food, or what it's made with. The 18 yr old High School student behind the counter cannot be trusted w/ understanding the severity of the situation. If you ever do fast food, check the websites of the place you want to go AND call their 1-800 number to check on the ingredients. For home cooking, it's probably best not to used peanut oil at all, anykind, so you don't take any chances.
Each consecutive exposure can cause worse reactions. Make sure you get an epi-pen - and as a pharmacist - get one from your pharmacy so you can personally check the expiration date and get a "fresh" one (they only last 1 year).
At the babysitter, if he is the only child there, then you will have to be sure that she understands he cannot have peanut butter, and peanut butter cannot be present when he is there - that she screens ALL his food (even those prepackaged gerber products) for peanuts - make sure she is clear that it is life threatening! If there are other kids at the babysitter, all their food will have to be peanut free.
Just b/c his frist reaction was "mild" - does not mean he will not have an anaphylacitc (sp) reaction in the future - leave an epi-pen w/ her and teach her how to use it.
If it were my child, the only way I would consider peanut oil is testing it at the doctor, in his office, w/ an epi-pen ready ---- honestly, if it were my child I wouldn't test peanut oil at all, I'd just avoid it at all costs.
My 3rd child was allergic to dairy (which he outgrew at about age 5) and my youngest is still allergic to dairy (outgrew egg, cinnamon, watermelon, and chocolate) so we spent a lot of time learning about food allergies and learned that we know many people w/ food allergies -- the peanut allergy trumps them all though - and we take that very seriously. We meet about once a week w/ a friend w/ a peanut allergy and we NEVER bring peanut prouducts around her! She is also allergic to egg, and cannot even be in the same room w/ an egg as the proteins sort of *waft* through the air and can cause a reaction.
I encourage you to learn all that you can about food allergies. FAAN is a great place to start! Eggs can be serious also, depending on what his number was on his test, but Peanut allergies are not to be taken lightly.
Good luck to you!