Im trying to bite my tongue (or fingers, damn technology screwing up good old sayings) and ive tried to be helpful, but you know, good old Aussie beach culture, I wouldnt have a problem with my 14yr old wearing a bikini of any description.
I think, in light of the last post, you need to pick your battles. Dont sweat over things like how long it took you to take the bikinis, if theyre against what you believe is appropriate, then the main thing is that she now knows. All teenager say no to just about everything, you could wave their favourite thing infront of them and half of them will say no just to be a pain in the ass, its that kind of rebellion that helps them develop, good old cognitive development. Dont sweat that kind of stuff.
Remember, when youre little, your parents know everything, when youre a teenager, they know nothing and when youre an adult you realise that they know about some thing and not others.
At the moment, as far as she's convinced, you know nothing, and that will change with time, but for now, she's acting like a pretty average teenage girl.