Your bank only need A CERTIFICATE OF RESIDENCE, that's proving both your address and the fact that you're here "long term".
You're on a ed visa so you can get this document easily from the immigration office. It will cost you 300 Thb there.
But prior to this, you need to make sure your landlord registered you online on the dedicated immigration website as his tenant (immigration will check on it and you can't get any certificate of residence until it's done).
At the immigration, you'll need 2 passport pictures, copy of your passport pic page, copy of your last stamp on the passport (related to your ed visa), and copy of your departure card ,
