Welcome!
I am also a professional with expertise in child development. I think you are likely right about your pediatrician. Our pediatrician told me frankly early on that he trusted that I would raise any developmental concerns, if I had them, so he was going to keep his questions on milestones pretty general.
As to cognitive assessment at barely two years old? I expect you know that assessments at this age are notoriously unstable. The best instrument developmentally for littles is the Bayley's, but the SB5, DAS-II, and WPPSI-IV also have norms down to age two or 2-6, and have the advantage of much higher ceilings. That being said, I don't know that a formal assessment at this age would be worth the money (out-of-pocket) you would have to spend on it, especially given the limited testability of tiny people. I typically recommend evaluation if there is 1) a problem being experienced by the child, for which more information may lead to solutions, or 2) if it may provide access to needed resources. Honestly, I don't see either test being met in your current situation, based on the data provided.
I hear your anxiety about her future, especially where there is a family history of mental illness in an identified GT individual, but really, you have time to work out school and peer relations. Keep educating yourself on GT resources and development, but even more, enjoy your lovely child, and continue to be present.
