This all seems pretty simple to me… you can’t start banning items of clothing. The burqa and niqab are clearly far more controversial and symbolically loaded than the average pair of socks or woolly sweater, but the wholesale ban of any garment would set a scary precedent.
It would also completely miss the point. The only reasonable objection to burqas involves the issue of oppression and female inequality. Fine, deal with that… question the values, not the symptoms. A total ban on the garments would just blur the distinctions between women’s reasons for wearing them, and mostly satisfies the sort of people who hate being confronted by difference of any kind.
The issue of banks and security-sensitive situations is something of a red herring. Just forbid ALL face coverings… motorbike helmets, balaklavas, whatever. Again, address the real issue, rather than hiding behind the simplistic prejudice.