I worked in a bank 40 years ago, so my experience is obviously a little dated.
The washrooms weren't open for the public to use. There is cash back there. Private financial information you may see on someone's desk as you go by and other security reasons. It was all about security.

Banks have long been targets of robberies. It was assumed that someone wanting to "case a joint" could ask to use the washroom to get a look and layout of behind the scenes.
In other words, there are valid security reasons why the public is not allowed to use the bank's washrooms. Even if you're a customer.

You could have walked in there wearing a five thousand dollar outfit and you wouldn't be allowed to use the washroom.

Now, banks being banks, if you had millions of dollars at the bank, and just walked out of a two hour meeting with some manager, the above would not apply. Then they'll bend over backwards to kiss your rear.