Claire, have you read all the “Letters and Papers” of Henry’s era that can be found online? If not, there could be something we miss.. I am hoping for a birthday present for Anne listed there, so that we can learn when she was born. Thank you for your answer!!
I have searched through Letters and Papers for mentions of gifts being given to Anne Boleyn, e.g. from Lady Lisle, but there is never any mention of it being her birthday and there isn't a particular date that keeps cropping up apart from, obviously, New Year when it was traditional to give gifts. Sorry!