It is possible for a child to use an alternative surname at school. A friend's legal surname was smith-jones (birthsurname-permanent carer's surname), but at school she just went by Jones.

Another mate went through all her schooling with surname Brown (her step-father's surname), only to find out when she was in her 20s and getting married and applying for a passport that her legal name was Green. Created a lot of difficulty because she had bank accounts, uni degree etc in the name Brown and couldn't just use her marriage certificate to change over to her married name Blue (because she had her legal name Green listed).

Messy, just messy.

So maybe don't use my first suggestion at all...
My schools dont use the name we keep telling them to, and its REALLY annoying the kids. On the form I write Angel Jones (Nee: Smith) - and they write everything out as Angel Smith. She hates it. Mr Smith signed away all his legal rights and responsibilities 10 years ago and has made no effort to initiate contact with the kids since - and they want to be Jones', like their mum, like the man who has been their father for just under 9 years, like their siblings. The schools refuse to play ball. its very disappointing for them.