Struggling Through This Issue of Modest Swimwear
To answer your question about why people can feel comfortable wearing a bikini on the beach but not their bra and underwear around town-- it is the fact that it is a social taboo everywhere except the beach. You'd be the only one walking around town half-naked, but on the beach you are among many others dressed the same, or skimpier, than you-- so you feel "safe".
Now as for Christian modesty, this is an issue that I've been struggling with. And, in many ways, I have to say that modesty can be dictated by cultural/situational circumstances. Just look at the "history of modesty" to see what I mean! By those standards women shouldn't wear shorts, short-sleeved T-shirts or even pants. But things have changed. It is no longer "socially taboo" to see a girl's arms or lower leg. How many men actually lust over a collar bone?
I don't see anything wrong with a one-piece swimsuit, tankini or possibly even a carefully selected bikini (one that actually covers your body, not held together with a string). It isn't just what a woman wears that "makes her sexy", but her attitude! You could have a self-confidant woman in a potato sack attracting more men than a woman who is wearing a bikini but hates her body. Use discretion. Don't flaunt your body, but "showing it off" (in a small sense) isn't necessarily bad. You can admire a man at the beach for his muscles without "lusting" after them-- why not the same for women?
I think we've gone overboard in our teachings. "Lust" isn't *just* thinking someone is attractive-- it is more than that. It isn't *just* picturing what a girl might look like with her clothes off. Can someone look at a painting of naked men and women without lusting? Of course they can! That isn't to say that people shouldn't use discretion when they dress. There are some things that are likely to "tempt" more than others, but it is important to remember that modesty is, at its core, more about ones attitude than it is about what they wear.
I'm not fully convinced either way, but for next summer I'm leaning towards a tankini for swimming and a bikini for when (if ever) I'm sunbathing at home. But I respect other people's decisions to use "stricter" types of bathing suits.