I too, have been with BofA for about 2 years already. They only ask you for your Consulate ID and I think, possibly a school ID. I have a Savings/Checking/ and a CD(certificate of deposit).

I asked them to input my ITIN on everything just for safe measure, I suggest you guys do the same if you haven't already. BTW, BOFA offers one of the better interest rates on CD accounts.