I will admit to having no experience on this question, but I suspect there are some folks in your local community who would have experience in this... Namely, your local triathlon retailer.
My sense is that a you may want to just go ahead and wear your sports bra during the race. I don't know how quickly it would dry, but hopefully it would dry quickly enough that you wouldn't notice it.
Another alternative is to simply put on your favorite sports bra after the swim in the changing tent between the swim and bike (T1)... Many triathlons will offer changing tents for just this reason.
Good luck!