L.R.
M. -- I wish I had references for you but I did want to say, if you get some, you may find your insurance requires your pediatician to refer you and you can't necessarily just book an appointment yourself without a referral (if you want insurance to cover it). Check with your insurance co. for a list of such specialists in the area who are part of your coverage, and take that list to your pediatrician ASAP. You don't want to go on your own to an allergist just to have them send you to a pulmonary specialist who then sends you to an ENT who sends you to someone else.
That said -- your pediatricians sound slack. Have they considered pneumonia? I've known kids and adults both this spring with coughing and wheezing who said "It's just allergies" when they actually had pneumonia that went undiagnosed. You put your concerns very well in your posting; if your child's having this much difficulty and labored breathing I'd call them, demand an appointment today, and go in and let them know exactly what you said here, ending with insisting they actually do something to help and refer you if necessary. They might be reluctant to refer to specialists or order tests because of insurance constraints. You may have to get very assertive with them. And if amoxycillin's not working, ask for other drug options--it can be the first drug but sometimes is just not strong enough. Good luck and don't let the doctor's dismiss your concerns.