Is dealing with banks, whether Islamic or non-Islamic, absolutely forbidden, even if the work involves non-usurious transactions, and should one leave such work to avoid doubtful matters?

It is permissible to deal with Islamic banks, unless the designed program involves a forbidden transaction. If the designer has doubts about dealing with these banks, he may refrain from doing so, to avoid scholarly disagreement. As for dealing with usurious banks in permissible transactions, that is subject to other fatwas.