"We strongly suspect that my 3yo son has a problem with milk and dairy."

 Definitely!!!  I bet you will be amazed at how the symptoms will clear up
once you keep him off dairy.  However, if after you have stayed away from
dairy, he still has problems, there might be other food allergies so you may
want to have him tested.  If our 8 year old has any dairy now, he gets a
clear runny nose--it's a dead give-away! And he also gets very crabby....