I agree with those posters who say that witholding privileges at 16 is not going to work, and that the way you need to tackle this is by having adult conversations with her.
The way I would start the conversation (or say that someone on Mumsnet has pointed out) is that this is actually conflict between her "personal expression" and the school's "personal expression". Organisations like schools and workplaces have "personal expression" too, and they demonstrate them by making rules. That is what differentiates the feel of school A where a straw hat and blazer is required from school B where torn jeans, purple hair and whatever else is fashionable at the time is what goes.
One may have strong views about various forms of this personal expression - from this is a ridiculous rule, to this is how we create a sense of community, just as individuals on this board have very different and strong views on piecings in general and nose piecings in particular.
But being part of an organisation involves sometimes having to give up one's personal expression, and conform to the personal expression of the organisation and if an individual won't, they will inevitably be thrown out of that organisation. That this is true of employers as well as schools, clubs, you name it. Suspension followed by exclusion from the organisation is therefore the logical and expected consequence of this kind of battle.
Sometimes the individual may, on moral grounds, need to fight the organisation where the rules are dangerous, or wrong - think for example of whistleblowers going against gagging rules to cover up wrongdoing. These people may be praiseworthy, but the fight takes an enormous personal toll. Sometimes we all agree (in retrospect), that the fight was morally correct. Think Nelson Mandela - but the consequence to him was that he was suspended from the society and locked up in prison for many years. The self expression of the country of South Africa, was, it now universally agreed, abhorent, and Nelson Mandela ended up president, so you can have a good outcome, but there were lots of other people involved in the fight for whom the outcome was not so good.
She as an almost adult, and one who has made a decision to get a nose piercing, needs to decide how abhorent to her an organisation that bans visible nose piecings while attending it is. If it is a tolerable rule, then she needs to flip up the nose piecing in order to conform, in order to get the other benefits of membership of this organisation, in this case, easy access to education (something that many women in many countries, even today, do not have). Alternatively, she needs during her suspension to research all available schools near where you live, to see if there is another one that she could apply for, and who would take her, with nose ring intact. E-schooling today is also a possibility,. She would need to look at the fact that such schools may well offer different GCSE boards, so the catching up at this point in her education might be difficult (or she might have to take a year out, and do her GCSEs a year after she intended).
You might also point out that as the organisation of "her family" you should have a say in that, as if she takes a year longer to do her GCSEs, you are likely to be funding her for a year longer than you anticipated, and it would not be unreasonable for you to ask her to get a job to pay rent to you for that extra year (at 16 she ought to be able to access a part time job as well, but again, employers offering jobs are organisations that have their own forms of personal expression, and that might include not allowing employees to have visible nose piecings). In which case, part of the research during her suspension probably ought to be into the availabiltiy of jobs in her local area.
So in terms of your original AIBU, yes are being unreasonable "To not want DD to get a suspension" - or, one can want unreasonable things, we do all the time, but you are being unreasonable to expect or hope school will not suspend. And if this is so important to your DD that she needs to carry it through, she will need to be talked through the consequences and the personal cost it will entail - and the only person who is in a position to do that properly is you.