I would listen to the doctor... she has been trained on what to look for, and will check on things you wouldn't even notice. The reading glasses force his eyes to focus differently, which would strengthen the eye muscle. He can still see it fine, but his eye would work harder and thus strengthening it. My little cousin had the same thing when he was 4. He had to wear the glasses for about 1.5 years (of course, at his age it was harder to keep them on...) It's especially important to fix any eye problems now, when the eyes are young and developing, than to let it become a permanant issue.
~after reading your update... I would still keep him wearing the glasses, but definitely schedule that 2nd opinion. My cousin didn't have Lazy Eye either... he had pretty much the same exact thing you described your son as having. One eye just didn't 'travel' as fast as the other, and didn't focus as quickly. The glasses DID fix the problem for him though... but yes, they gave him headaches at first too. I get headaches when changing my prescription glasses as well, so it's to be expected when making a visual change. Once he gets used to them, they will stop giving him headaches.