I carry an Epi-pen. I take all reactions seriously. A mild reaction this time, doesn't mean next time will be just a mild. I learned, from my own life threatening experience, how 1st occurrence mild to 2nd being severe, how serious reactions can be.
Allergy testing is a must. They'll only be able to do a blood test on an infant, not patch testing. Results are not 100%. False positives are common, and so are false negatives.
Remove all peanut products from the house. It's the easiest and safest way to eliminate allergic reactions by contact.