I am not anti-skip in every situation; rather, to me the OP's set of facts weighs in favor of sticking with the awesome teacher and giving the STEM school a chance to get their 5th grade program together. The OP's DD only has one more year of elementary school (and with an awesome teacher) since her middle school apparently starts with 5th grade.

You don't need to be in middle school to compete in middle school competitions or in high school to compete in high school competitions. DS, with one exception, has always competed on mixed school teams for math competitions. One of DS' teammates on a couple of his mixed school teams is a 5th grader who has excelled in many math competitions, including scoring decently on AIME.