I have a couple vine maple bows like that and I had some success steam heating the bent area and bending the other way. But I didn't care if I destroyed the bow in the process. I steamed for an hour then quickly clamped it on edge in a stave press and hung about 40 lbs. of weight from the end of the limb, bending it slightly further than I intended, knowing that it would spring back some in the next 24 hrs.
Its hard to bend the bow laterally like that and I know guys who would bend the handle instead. Or you could do a little of both. Depends on the wood, depends on the size of the bow, etc.
It is a little puzzling that yours is so bad that the string pops off yet the string falls near the handle anyway.....?