I am SRRV and do not have an ACR card. I had no issue opening a bank account or getting a lease.
I did have to explain to Globe that I was ACR card exempt, then to a Globe manager the same thing, to get my internet account approved. Once the manager saw that I was SRRV he OKed the application without having the ACR. This was despite it saying on my SRRV card that I am ACR exempt.
I would guess that a BB visa is similar, just explain 3 or 4 times politely that you are ACR card exempt and do not have one. Then escalate the issue and politely ask for a manager. Explain twice more then go to the bank next door where they might know the rules.
This of course will be easier in a place with lots of expats where they have seen the various types of visas. Just remember bank clerks and managers are not immigration experts and when they see a spot on the form asking for ACR card or other information blindly want that spot filled in.