Welcome to the board! I hope you stay and post often
I agree that you have great experience that will go over well in a high school setting. In my experience, high school administrators most certainly hire specialists (like social workers or counselors or school psychs) from elementary and/or middle schools. The kind of training those positions get is usually not as specialized for an age level as teacher training, and the certifications for those positions are usually K-12.
I guess what I would do in your position would be to see if you can learn what the school needs/what struggles a particular school has, and highlight the skills/experience you have that would benefit the school in your cover letter (without pointing out that they have these deficits, of course). A solid cover letter is hopefully what will get you interviews, and then you can sell yourself in person. Good luck!!