When a birth is registered does the child's surname have to be that of its mother or father? Can the child's parents give the child a completely different surname if they wanted to? Not sure why anyone would want to do this - just another hypothetical question that arose out of the usual office banter today!
I had a look at my own birth certificate (rather ancient) and it doesn't specify 'my' surname at all, just first name. Mrs dzug's, of around the same era, is the same.
The I looked at our children's - rather more recent, and it does record surname as well. So procedures have changed since nineteen mumble mumble.
Given this I'd suggest Hawkwalk is right and you can choose the surname.