Dd is high attaining but underachieving. Suspect adhd. What next?

[BigBirthdayGloom]

Dd has just had her year three report. On the face of it her attainment is great-3a teacher assessments across the board with 4c test scores for reading and maths and 4c teacher assessment for reading. And yet, her teacher believes her to be under achieving considerably because she inconsistently concentrates and her work output is lower than he'd hoped. Plus, she struggles to listen to her peers and this is starting to affect their relationships.
I have been referred for assessment for adhd and I tick all the symptoms with a catalogue of examples. I idly looked at it for dd and was shocked that she seems to fit the profile too-not just the obvious things but the more subtle things too.
How far will I get with requesting that dd be assessed too, given that she isn't underachieving for her age by any means but has had reports suggesting she's underperforming for her since she started school? And is there an advantage to going private for the assessment?