There are bras available that have a "soft underwire" ... it's not really wire but a flat plastic. If the bra is causing so much discomfort I would go shopping until I found a bra with enough support but without a wire. I realize that you haven't had any type of breast surgery and I don't know how much you usually pay for a bra but you might look into mastectomy bras. They are quite a bit more supportive than any regular bra and also wear a lot longer. The fact that each cup contains the extra piece of material to form a pocket wouldn't make any difference. They aren't really any more expensive than a "good" bra .. it's just a thought.
The problem with underwire bras is in a case to case basis. Some women will have no problems with underwire bras, while some will have pressure problems or irritation problems on the skin in contact with the underwire. The tightness of the bra will also come into question. Increased tightness may cause local irritation of the skin and can cause pain. This may also affect some nerve and blood supply of the breast causing several problems and all sorts of pain.
Be sure to have right amount of tightness to avoid irritation of the underlying breast skin.
Regards.