Could you go in for a couple of hours on the second Monday so that it doesn't count as a full two weeks? Instead it, would be 5 days the first week + 4 days the second? Missing 9 days is much better than missing 14. I also don't understand the policy of counting the weekend days - that seems insane.
Do you have a teacher's union that can advocate for you? Can you get put on disability for the two weeks, instead of taking them as sick time? even if the surgery is elective, that may still be an option. Ask your doctor and/or union rep.
Do they have to know that the surgery is elective? I mean, if it's a boob job, ok, but maybe you could find a medical reason for a different surgery? Or just not tell them the specifics at all - just say you are having surgery and the doctor is requiring two weeks without work.
Also, why can't you do the surgery during your 30 days of vacation? I know it would suck to take up half your summer recovering, but it might be better than using up all your sick days on student-free days.