I think you have to be accepted by the school first, then sort the permit out. But it’s worth talking to your gemeinde to see what they say. If they say no to an extension though, don’t overstay. It’ll only cause trouble for you down the line if you want to visit/live/work in Switzerland in future.
I'm planning on doing something similar (switching my L permit to a B study permit), but I will also be leaving the canton of Zürich to go to Geneva, which makes the process a bit more complicated!
In any case, I called the cantonal authorities - in your case, I think you might have to call the Zürich Migrationsamt (or, if that is wrong, they will tell you who to contact). Hopefully you will be able to apply before your current permit expires, otherwise it will cost more and there is a but more paperwork involved.
Also, as someone else mentioned, you have to be accepted before you can apply for the visa - or at least have proof of contact in some form.
Good luck and keep up updated, I think there are quite a few people who would be interested to know how this process works!
I got the extension of 6 months cause I have been working in Switzerland for more than 12 months, although the music school is not full time it said it cannot be counted as official education. But I applied at another College and its full-time. so that counts!
Thanks for your help everyone, I hope this information will be useful to you