I hesitate to put this forward as a suggestion, as a solid diagnosis really requires quite a bit more than just a handful of rating scales, but if this is truly of value to you, I think the direction the contract psych at the school is indicating is actually to have his primary care physician originate the Vanderbilt ADHD Rating Scales. That should cost you no more than whatever your office visit co-pay is, and would put the rating scale information in the hands of the medical professional who would be principally responsible for managing ADHD, should that be relevant.

I'm more interested in whether there are IRL functional concerns, other than being a bit bouncy and impatient with lengthy directions, or if he feels any distress about his own attention and activity level, independent of impressions he may have picked up from adults. Having ADHD-ish qualities (even ones that meet many diagnostic criteria) does not necessarily equate to pathology. FWIW, we have one with many similar characteristics. As it happens, I have access to many of the rating scales used for ADHD, and ran through the items one day on my own child, for my own interest. Not at all to my surprise, the scales indicated a fairly high likelihood of ADHD. That has not, however, translated into functional impairments. Having small classes (when in school), with at least moderate academic challenge, and teachers willing to view the profile as charming and fun, rather than disruptive and distractible, have all been significant factors in outcomes for this child, while we worked together to develop EF skills and compensatory strategies. (Which, of course, we all need to work on, to varying extents, with our children.)

In the case of your child, I would probably start from examining the appropriateness of his level of instructional challenge (paradoxically perhaps too low, if he feels he can put the bare minimum of effort into tasks), and possibly have the 3rd grade team (who didn't view him as disordered) pass some insight into effective teaching strategies for him on up to the 4th grade team.

Bottom line: if it ain't broke, don't fix it.