Bras that are absolutely 100% cotton are hard to find. I'm able to wear Fruit of the Loom brand sports bras. They are inexpensive, so if they don't work for you, it's not a huge loss. I don't know if the thread is cotton or not; it might or not be. I guess you could contact the manufacturer to find out. I assume a big company like that has a consumer relations department. I don't break out when I wear these bras, but then it tends to be the elastic that I am allergic to, rather than the thread. The elastic of these bras is totally covered with cotton fabric. I have read suggestions elsewhere to wash new garments several times (in detergent that you tolerate, of course),because sometimes it is the finishing chemicals on the garment, rather than the textiles themselves, that are irritating to the skin. Unfortunately, a lot of sellers aren't very helpful on questions like this, because they have been "burned" by customers who wanted to hold them responsible when the garment caused a rash. For that reason, sellers may not to want to make any claims that a garment is hypoallergenic, even when it is. So maybe the best route is to try to find something that at least looks like it would work, preferably from a major manufacturer, and then contact the manufacturer to inquire about the materials. Good luck.
I'm a guy so I have no personal experience, well I suppose I wear undergarments and outer-garments that have synthetic thread and it touches my skin... seems I am not allergic I now conclude.
How about buying "100%" cotton - here I assume no elastic and that is possible. Seems to me if they advertize 100% it should be 100%, not 99.9% with 0.1% thread of other materials.
Good luck, maybe some cotton adhesive tape in critical areas... if that can be had with no synthetic materials... don't know that either.