Honestly OP, I don't think wearing an underwired costume for lane swimming is going to work. It might if you're doing just breast stoke, but as soon as you start doing crawl, the side-to-side motion will make the wires dig in (trust me on this; I've been a regular serious swimmer for many years, and I've tried lots of different types of costumes).
Fashion swimsuits are no use for serious swimming - you need a sports swimsuit. Have you tried a basic, classic-style Speedo racer back? I think that might work for you.
On the subject of tankinis being no use for lane swimming ... the fashion ones won't be, but if you can find a proper sports one, it will be fine. I've got a very solid and supportive Speedo tankini that's great (the only problem with it is that because the material is so think and it fits so closely, the top is a devil to get off).
The suggestion above is good - wear a sports bra underneath if the costume doesn't seem supportive enough. Also, you could try a costume with a rash vest over the top for a bit more support.
Another suggestion for you - try outdoor cold-water swimming. At the end of last year, after an entire summer of daily swimming, I mysteriously dropped a few pounds in October/November when the temperature of the pool fell - I'm convinced it was because my body was having to work so hard to keep warm that I was burning extra calories.