Immigration issues a certificate of residence but some embassies issues an equivalent to that certificate. Thats the one the bank asked for. You can ask them if immigrations certificate will suffice.
Some embassies issues an equivalent to immigrations certificate of residence and there are banks that still asks for that certificate. Years ago Bangkok Bank in Kanchanaburi required that certificate. I don't know if they still do. One of Bangkok Banks official requirements (until a few years ago), was that certificate or immigrations certificate of residence.
You can not have the money required at immigration split up between 2 banks. At least, I have never heard of that before. What did your parents know about Thai banks when you were in your teens?