Yes, I believe you can IF the circumstances for being signed off are workplace-specific.
In that case, you would be signed off from the specific job you are doing in a specific working environment, not signed off from practising your profession altogether. There is a difference.
A friend was recently signed off sick and the doctor wrote the reason on the form - it was due to bullying from senior management. This is not a circumstance that could be repeated in another workplace, even within the same profession, as that same senior manager would not be working there as well.
As long as you stick to doing supply on your non-contracted days, I don't see that the school would legally have a leg to stand on if indeed your sick leave was due to bullying or another school-specific problem.
This being said, I don't think your school would look favourably on your decision and may make your life harder if they found out. You need to decide if it's worth the potential hassle of having to fight your corner and create more ill-will. If the job is making you ill, you'd be better off resigning now, negotiating early release and getting a new job for a January start.
Whatever you do, seek union advice before making any decisions.