Sadly if the mother refuses, the CSA tend to not do anything these days. In the past they used to stop claims and infer you are not the father. But if someone is on the birth certificate as a default they assume he is the father, and would keep assuming that if he declined a DNA test. Double standards, perhaps.
Whilst a private DNA test wouldn't be legally binding, it would indicate if it's worth going down the formal route, so do consider that if there's enough contact to do so.
Next steps would probably be better with legal advice and a solicitor.
On parentage you would need to apply for a Declaration of Parentage through the Family Court.
If access/contact arrangements are being changed, then that needs a court application for a Child Arrangements Order. (or use Form C100). Mediation may be recommended first though.
So, searching for a good local Family Law solicitor may be easiest as a first step.
Also you (or a solicitor) could write a formal letter to the school's Headteacher. State clearly that you have PR, you do not consent to a name change, and request that the child be referred to by her legal surname in all formal documentation and registers. However, do also think (particularly on the name) about the relationship with the child, and how much she wants to do that.
This site might also be useful: Family Law Questions? Ask a Family Law Solicitor for answers ASAP