Or give your boss a raise...

The difference with this situation is that a snapped bra-wire constantly stabs you, which actively hurts. Good luck, OP, for me it helped to put a bandaid or some tape around the pointy end of the wire and try to make it stick to my bra instead of stabbing me

You can leave a workout and come back whenever- you don't have to get permission to leave a workout. However, it's highly unlikely that that OP's employer would let her leave work to go home, take off the old one, put on a new one, and come all the way back to work. Especially if OP's manager or boss is a man- and before someone tries to jump down my throat about being sexist or whatever term you want to use, I'm saying this from first hand observation- because men without a wife/girlfriend, or a mother or sister they are very close to, don't quite understand that a snapped underwire isn't like, say, a broken zipper. It isn't a matter of it just being uncomfortable or increasing the risk of a wardrobe malfunction- snapped undersides can be flat out dangerous. They can cut the person the bra is on, or the wire can even act like a needle and thread and go in one spot, come out the other, and be laced under the person's skin and even muscle. It isn't likely, but it can happen, especially if the job is physical labor/movement intensive. (Which, yes, I have seen this in person. Not pleasant to look at, I don't want to think about how much less pleasant it was for the girl it happened to.) Note that I'm not saying that male employers with that view on them are being intentionally jerks about it, but that they don't know that broken wires are dangerous.