This is exactly why they repeatedly and emphatically encourage candidates to do the CFA early; even if at least to get a minimally qualifying score in, and then maybe take it again later. My son made a similar mistake last year. he trained all summer, and got himself into great shape, but he wanted to push a bit farther so he could get a perfect score. He was working on the shuttle run and needed about 5 more feet for the max on the BB throw. Unfortunately, he injured his shoulder and clavicle in soccer, and was unable to do the CFA, or even train. Ultimately he did take it before the deadline, but he still wasn't fully recovered and he was in pain; particularly with the pull-ups, push-ups, and BB throw. he got a very good score, maxing most, so all ended well, but that is just the way it goes.
I think it is a set deadline, without exceptions, but it may be different for each academy. My son was doing it for USMA, USAFA, and USNA.