When you fast you do continue to have your period. Some people actually have prolonged or heavier periods while fasting. I have heard a few people say that they had lighter periods, not often. Mine were the same flow, but a day or two longer than normal. I do have very short periods anyway though.
As far as staying on the fast, I look at it this way; If I am going to be miserable anyway I might as well be doing something good for myself while I am.