My son was prescribed his first pen at 5mo of age - weighed about 12lb, allergist always said that the good it could do would outweigh the risks. Now this particular allergist was an idiot and discounted DS1's barley reaction as ana (though hospital disagreed) and told me when I called that only nuts would cause an ana rxn.
DS1 tests positive for a host of things, had eczema, etc. Honestly given current guidelines, he would have never been tested for many of these things (top 8 spt). At his initial test he was positive for wheat, egg, soy, peanuts and the tree nut mix. Now he may have outgrown soy but current allergist given his long list of false positives in the past that he was never allergic to peanuts and tree nuts (this is the kid who tests positive to rice, corn, oats, soy, dairy, peanuts and tree nuts but is only allergic to wheat, rye, barley and egg).
Well it's nice in theory to just avoid those allergens, realistically that baby will be crawling and walking soon with likely no issue with putting random things in his mouth. Honestly, I would see if the ped would prescribe and epi or find another allergist. I know our current allergist would prescribe an epi.