According to our pediatric allergist, you can do allergy testing on a newborn (but probably don't need/want to). Our son was allergy tested at 18 months (the "scratch tests). Keep asking your doc to give you a referral to an allergist--these kinds of allergies don't get better with time/exposure, and usually only get worse. Allergy testing will help to
My brother & SIL had a large dog, whom passed away when X was 1 and Y was 3. Neither X nor Y ever showed any allergy symptoms (that they noticed-no itchy eyes/congestion/etc). About a year later, my other brother got a puppy. The puppy, being a puppy, licked X's face, including her eye (ew). Her whole face swelled, got red and her eye itself swelled (yikes).... that's how they found out that X had an allergy to dogs.... Apparently puppies are more allergenic than adult dogs.
Zyrtec and Singulair are 2 medications that can be prescribed for children as young as 6 months, I believe. Ask your doc about those. And definitely keep pestering the doc for a referral to an allergist. Benadryl, of course, also works, but shouldn't be given w/o first talking to the doctor.
In the meantime do limit the exposure to the dog, and make sure to wash his hands and face as soon as possible after exposure to the dog. (no petting or giving treats or helping feed them--no letting him lay or sit in or near where the dog lays/sits). Bathing the dog regularly, washing the dog's bed regularly, and using a Swiffer (or similar) to keep pet hair cleaned up may also help.