You should give this serious thought as any name change OTHER than adoption will not be recognised by the Phiippines Embassy. So if you go for British Citizenship, that will be in the new surname, as will a British Passport. BUT if you go for Dual citizenship the new Philippine passport and oath taking will be using the OLD surname. Could get complicated!
Adoption is long-winded but costs £170 I believe and is recognised by both countries and also means that in the event of both you and our wife dying then your step daughter would inherit your estate. Also no problems with being a beneficiary of your pension scheme if you're in one. You do NOT have parental responsibility for your step daughter unless you adopt, get a Parental Responsibility Order, Residence Order or Special Guardianship