Of course, there's only any point to this if you commit to being as honest as possible with the assessor giving the second opinion. I know how easy it can be to (even unintentionally) skew an assessment one way or another (though with an experienced and skilled assessor, they should be able to minimise the impact of this), through things like your choice of assessor/assessment service, the information and documents you choose to present or withhold, the way you frame your descriptions and the language you use to characterise the difficulties, emphasising or de-emphasising different aspects, arranging or not arranging things so that the person being assessed is comfortable and has any supports they need, and so on.
I'm not insinuating that you're a dishonest person, by the way — I think it's human nature to present things in the way that's most likely to get you the outcome you really want or that you believe to be the correct one, even if you try to be scrupulously honest.
For me personally, I have (NHS, adult) diagnoses of both ASD and ADHD, and for both of them, I know that the mindset I approached the assessments with, and hence the information the assessor had to go on, may very well have had some influence on the outcome.
WRT ASD, I've been told by an autism-experienced psychologist who knew me very well that she had no doubt I could "fake normal" in an assessment if that's what I wanted to do — I'm sure that could be subconscious, too, if I really believed I wasn't autistic and/or didn't want a diagnosis.
And for the ADHD in particular, I don't actually think of myself as having ADHD and never have, in that I don't believe that the difficulties I experience are truly those that were intended to be captured by the ADHD diagnosis criteria. I do technically meet the criteria, and at no point did I lie to or try to mislead the psychiatrist assessing me, but for the ADHD assessment I was aware I was framing my difficulties differently from how I usually do — through an ADHD lens, rather than an ASD one. I was invested in obtaining the diagnosis because I suspected the ADHD medications could help with what I think of as my ASD-related executive functioning and circadian rhythm difficulties, and this preferred outcome inevitably influenced how I presented things to the assessor.
I wonder if you were motivated to get your DD assessed by something like a hope that medication would help her, or by a desire for a no-fault explanation with a ready-made set of strategies to try, and, looking back on the assessment and diagnosis process, feel that, at that time, you may have presented things in a way that skewed the picture the assessor received — and therefore now feel unable to trust the outcome. Unless the assessor was very poor, they shouldn't have given a diagnosis based solely on symptoms displayed during a recent period of unusual stress, so it's likely that there were other aspects of how your DD is that, when presented in the right way and examined by someone looking for ADHD features, would fit that framework, but perhaps you're now looking at those features through a different lens. (And if it was a poor or inadequate assessment — and other posters have pointed out that school input would normally be taken into account — then another, better assessment would seem to be in order anyway.)
Of course, the risk of your own hopes, desires and beliefs affecting the outcome of an assessment would exist with a second one, too — if your feeling now is that your DD does not truly have ADHD, the same factors could apply. There's also the risk of ending up feeling even more uncertain, if there's one "yes" and one "no". That's why it's vital IMO that you commit to giving the assessor access to all relevant information possible, including the previous assessment report and if possible information from school and any other relevant settings. And it's important to go into it with an understanding that the second assessment may confirm the diagnosis.
It seems to me that, unless you get a second, as rigorous as possible assessment, your uncertainty over this diagnosis, and unwillingness to disclose it to people who need to know, will hang over you and your DD. If she has a reliable diagnosis of ADHD she has a right to know about it and she shouldn't be expected to keep that a secret if she doesn't want to. If you feel her diagnosis was unreliable (either because of an inadequate assessment, or because of the natural human tendency to present things in a way that will get the outcome we believe or want to be right, or both) then you shouldn't leave her with either (a) a question mark over the diagnosis of a condition she genuinely has, or (b) an incorrect diagnosis which hasn't been overturned — and shouldn't burden her with having to keep it from everyone at school, either.
Either she has ADHD, and your uncertainty about the correctness of her diagnosis and your unwillingness to disclose it will mean she doesn't receive the support she should. And there's the fact that either you keep her in the dark about it, or make her feel she has to hide it at school. It also means she can't feel confident herself about her diagnosis, or validated, or any of the other emotional purposes a firm diagnosis can serve.
Or she doesn't have ADHD, and your decision to get her assessed, plus your choice of assessor and/or decisions about what information to provide them with have left her with an inaccurate and uncontested diagnosis in her past and on her records, which could detrimentally affect her in a whole different set of ways.
With the best of intentions, you caused this situation, and the only way I can see of fixing it is to reassess with extreme thoroughness. Ideally NHS (they're still seen as more credible by some), but we all know she'll be doing her A levels by the time that comes through, if they even let her on the waiting list in the first place