I'm in a hurry so sorry this is rushed:
My daughter is bright, sensible, caring, organised just like yours. But a little younger.
She hates the idea of growing up, won't admit that her periods have started or that they happen relatively regularly, would rather hide the evidence all over her room than acknowledge it.
The problem with my daughter is not with me...at least after much soul searching i don;t think so. We still talk about everything else except growing into a woman...I have always been open about periods etc as well as relationships sex and the like. We could always talk about it...until periods and breasts actually happened
With my daughter the problem is with her...she does not want to acknowledge it is happening. She would rather lie (and generally she has a real problem with lying) than admit periods happen to her. She can't even bit the stuff, she would have to admit to herself she was hiding something...and then there would be no point...she would have acknowledged it.
I have tried (not only once and not in this order) direct confrontation (tears and anger), subtle information (printed and letters), ignoring with sanitary products around for use if and when (not used...loo roll instead). Her younger sisters are easy and open, as a family we have plenty of female role models who are open and relaxed about sharing.
I am just sharing this to let you know you are not alone.
Lying to the medics would be because she is lying to herself.
For my daughter I think the school education system of doing development and puberty related to sex and relationships messed up her head and emotions about this.
HTH
Best wishes xx fool