Different sport, but testing became a major PITA for us. Not the actual testing so much, but the scheduling. Between the HS season and off season competitions and camps, there were really only 1-2 dates a year that didn’t cause some difficulties. We had a pretty robust off season schedule, but it was also a bit random. He might have one weekend that month with a tournament, but if that was testing day he was out of luck.
TLDR: take it as soon as you can, even if not fully prepped, just to have SOMETHING on the books you can use if needed. Also it sounds like based on PSAT scores a SAT taken with no prep will still probably be enough that a coach is comfortable knowing that your athlete is recruitable. They don’t want to waste time just to find out that the test ends up in a spot that the admissions committee would give a hard no on admitting the student.