The reason the oil might be ok is because it the proteins/ amino acids in foods that cause reactions and oils are often striped of proteins. Peanuts have a lot of different proteins, and its almost impossible to pinpoint the exact one or ones that are causing the allergy. This is why immunotherapy is not done for peanuts or any allergy. Also, it tends to not work.
I woundn't eat peanut oil though, and due to the increase of peanut allergies and the litigious society we live, I highly doubt any major fast food chains are using peanut oil anymore. Though it doesn't hurt to ask.
I'm allergic to peanuts and the allergy only started in the last 3 years. I've been tested to confirm this, and I have mild allergies to many other tree nuts, to many fruits in raw form.
I understand that I'm an adult and more able to avoid food I cannot eat. I guess the only solution for a child would be to make a rule that they can only eat what you allow them them to eat, and as they get older teach them to read labels so that they will be able to make safe food choices as they get older. Again with allergies on the rise, many food will list common allergens seperately and in plain language and will tell you if the foods were made somewhere where they could have come in contact with allergens. Also make sure to periodically teach them about Benedryl dosing and how to use to the epipen.