I doubt you're thick, but yes, afaik you need her FULL birth certificate - not just the short one, saying when and where she was born, but the A4 one saying who her parents are too. This will have been available to purchase when you registered her in the first place, and presumably can also be bought, from the registry office, retrospectively.
Your counter-signatory will probably also be contacted, perhaps at work, to confirm what they have said, i believe. Good luck. It's a pain jumping through these administrative hoops!