My dd has food allergies. For the previous posters, coughing can be associated with food allergies. FYI also, lactose issues are an intollerance, not an allergy. Intolerances can cause lots of issues but ultimately are not life threatening. Food allergies are potentially life threatening. I'd contact another ped, or allergist or gastro. Maybe asthma? Allergists do that too. RSV may also be a concern. Not sure who to see for that. I think you can call Children's Memorial 800-Kids-Doc and they may be able to tell you who to see. Good luck!
ETA: the loose stools and spitting up are more of a concern for allergies for me. Maybe the hypoallergenic formula still has proteins in it of the allergy or he has more than one allergy.