There are a number of possibilities when dealing with banks who don’t like CMRA addresses (that is what the US Postal Service calls mailboxes provided by third parties, such as UPS, Mailboxes Etc, and L4SB).
First, remember that if you use a CMRA address for your company (like the L4SB Virtual Mailbox), it’s the one address used on the Articles to help protect your identity. That doesn’t mean you’re stuck with using just that address for any and all business. Oftentimes, banks want a physical address where business will actually be conducted in the state where you are located. That can be a problem, if you’re trying to use a L4SB Virtual Mailbox that’s located in NM, when you’re located in Florida (for example). The answer here is, to give the bank your actual address where you’re conducting business locally. The bank will keep this information anonymous, so it won’t be published on the Internet.
Second, if the bank won’t let you use multiple addresses for your business, then we recommend trying a different bank.
At the end of the day, banks have different rules around this, and those rules will be inconsistently applied to folks, depending on whether the bank has a pre-existing relationship, and other rules (national and local) related to your type of business. Some banks are hostile to certain types of businesses, and other banks may be more accommodating. It’s something you need to shop around, if a bank is going to give you trouble with something as mundane as an address.