Not unless the bank allows unsecured connections in the first place. What bank does that?
The feature in question forces an encrypted HTTPS connection when you attempt an unencrypted HTTP connection, thus sparing you an error message. But the absence of that cookie will not get you an unencrypted connection if the server does not allow it.
Good question. . . I sincerely hope that there are none out that will. . . but it was a scenario that was brought up in discussion of the pros and cons of the Super Cookie's uses. . . and what could happen if you delete them and why you may not want to. Once you've deleted the cookie, it is gone, even for the next session when you are NOT in Privacy Browsing mode. . . along with the data the bank may need, requiring you to input more information each time you log in.