To be fair, OP said earlier that the assessment included input from her DD's tutor, which isn't really the same thing as school involvement but is at least some kind of external perspective from someone who knows her within a different kind of relationship, albeit from a person who probably sees the DD in a home environment (and in any case is directly paid by the OP and may have been reluctant to contradict her). I wouldn't say it's a reliable diagnosis, but it's at least suggestivez and I wouldn't think it's safe to just ignore it.
Given what we know — past diagnosis of ADHD, possibly inadequate assessment, borderline test results, reduction in symptoms after temporary stressors removed, OP's doubts about the diagnosis — it would, IMO, be irresponsible at this point to either disregard the earlier assessment altogether and assume it was a misdiagnosis, or to go full steam ahead and treat it as a 100% sure thing. And in either of those cases, keeping it from the school would put stress on the DD (assuming she knows) and could backfire badly.
To me, the only sensible thing to do is to tell the school everything — the reason for originally seeking assessment, the existing diagnosis, the reason for doubt over its correctness and the DD's current functioning — inform them that you're planning a reassessment with a reputable multidisciplinary private service, and say that you'd appreciate their input when the time comes, after they've had long enough to be able to offer an opinion. Then go to the second assessment, provide everything to the assessors including the previous assessment report, try to be honest and not talk anything up or down, and see what happens.
You can't just leave a diagnosis of a neurodevelopmental disorder that you've lost confidence in wafting around in the air like that, not knowing how to answer questions about disabilities when either answer feels like an untruth.