I don't have any specific experience with this, but here's something to consider: Unless they're just not paying attention, the bank will know it's an ITIN, and not an SSN (or EIN). This means they will know that the applicant is not eligible to work in the United States (including self employment). What they do with that information, and how it affects the overall application and/or how other information provided on the application is interpreted (such as employment and income), probably varies by bank.

That's definitely a question for a lawyer, CPA, or enrolled agent, and I'm none of those. Having said that, I'd be very surprised to find that having or not having an ITIN affects either one's obligation to pay US taxes, or obligation to file a US tax return.