This may sound like a crazy idea, but could it, perhaps, be an interesting experiment to spoof your user-agent string? You might, for instance, want to make it look like you’re running Firefox under Windows, and see if that allows you to log in to the bank site.
(To be honest, I’m generally strongly opposed to lying about your system’s identity like this, but if the bank site does let you in when you do this, then the bank does something awfully stupid in its login procedure. In that case, it’s time to send a complaint to them—I had to do that some 15 years ago... after which they invited me to test their new developments before they rolled it out generally—How time flies! By the way, do remember to undo the spoofing as soon as you possibly can!)
I’m not sure how exactly you would configure the spoofing (or even if recent Firefox versions still support it in the first place), so—assuming that you do decide to try this out—you will have to look it up.
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