My bank knows me well enough that I can use the drive up window in the dead of winter while wearing my balaclava and goggles. Might have something to do with me knowing one of the bank managers pretty well because I live next door to her elderly parents and help them out with yard work and such.
But the way I see it, a bank is a privately funded company (read: non tax dollars), if they don't want bikes in the drive thru lane they have every right. But banks make money off your money, and if they want to make it so hard for you to give them your money then you can find someone else.