You have posted very sensitively - well done for thinking through how to challenge this - it is a typical tactic to say ‘oh this is not important’ so please comply… all it does is lead on to other expectations…
there is quite a lot written on here and elsewhere about why pronouns are important, and we should be accurate, so you are right to be concerned.
I would agree that the name is easy and anyone might choose to shorten their name, but I would be saying to the head that you can not support the concept of non-binary which is meaningless, at a school you expect fact to be taught - the girl is of an age where there will be sex education etc and that will teach that we are male or female, it is not reasonable to expect that teaching to be corrupted to include imaginary concepts such as non-binary which are biologically impossible…
so the head will need to decide how to deal with the parents when those lessons come up
the head should also be reminded that non-binary has no legal meaning and that being gender critical is a legally protected characteristic…
the head needs to be a bit more assertive with the parents and tell them that name shortening is not a problem, but their belief that their daughter is somehow neither male nor female (when they are the only two possibilities) is not a concept that will be supported in this school - to support it would be to teach it as reality to all other pupils, and as well as being a lie, those pupils and their parents will also have views and in supporting the one child the head is likely to be upsetting many more…
the parents have a choice, if they don’t like a school where truth is taught they can choose to go elsewhere…
as for comments above about not abusing power - ignore them, there is a specific agenda there… your power as an adult and staff member is to guide and help children grow to become adults, not based on perverted distortions of the truth, but based on reality, fact, science, truth. To do so is your job, not an abuse of power.