Go to a different bank. You can also remove a credit freeze same day and then reapply it the next day. Last time I applied for a credit card(for the sign up bonus) I was able to do that.
This is strange. I could understand them putting a long hold(7-10 business days) on the deposit since it's a new and a large deposit, but not letting you open the account?
Possible logical explanation: The banker used the wrong terminology. They weren't going to run a full credit check, but they might use the credit agencies for identity verification purposes. I use to work in a US based bank years ago. When I opened a new account for a brand new client our computer system would verify your SSN matched your current address, last address, full name, drivers license number, etc. The system didn't run a full credit check, but I believe it collected all of that information from the credit agencies.
If someone didn't pass that test, for instance due to a frozen credit file, then we could still open the account the old school way. In addition to your government issued ID, you would need to show us your social security card, and 2 bills(utility, cable, credit card, etc.) with your name and address on them.