Overthrow the Queen, or force her hand. She's not willing to listen to reason and there's evidence those above her are.

Go to local media outlets with your story with plenty of spin (like your post! But no dragons) Garner public sympathy and point the finger squarely on the Queen. Emphasize the current crappy state of affairs and the embarrassment of the existing website. Emphasize the learning opportunities for the kids. Try to also get articles in school bulletins and publications also (make it less political though). In that vain you may want to also host it publicly as previously suggested by a few others.

Pester the higher ups that the Queen is inhibiting the ability of kids to keep up with the "technological progress of the 21st century." Go to school board meetings persistently to apply pressure. Use every tool at your disposal.

Distribute a petition around school and get a bunch of signatures. Write letters and meet with your congressperson. Garner allies on the board and at your school. Post your story on Reddit!

Basically, make it more of a headache to the Queen to deny your best intentions than to "take the risk" of relinquishing control of the website. You've come a long way with the programming and technology, now it's time to turn your attention to the political state of affairs.