Even with a long assessment appointment, I wouldn't necessarily rely on a regular pediatrician to diagnose a 2E child. The giftedness can mask so many other issues and pediatricians are (probably) more attuned to obvious cases. For us, the most important specialist was a neuropsychologist. Even then, though, it seems like there are so many overlapping symptoms that it can be difficult to tease out an accurate diagnosis.
It sounds like you have a good handle on the most pressing issues at this point, with the motor hyperactivity and social issues. The tricky part is figuring out what's causing them so you can use appropriate interventions. Can't imagine any harm done by social skills training, at any rate. I think (but am not certain) that OT would be helpful if there are specific physical skills that need work. I'd want to know more about the diagnosis before spending a lot of time (and money) on therapies.
