Update on this post.
I just got off the phone with a CIBC rep. They're accepting my pay stubs and a new card will be coming in the mail in the next week or so. They said they'll be talking with the other rep I dealt with, and will review their policy on private information.
But the best part was that they contacted ME. Apparently, they found my screed on this humble blog and wanted to rectify the situation.
I knew this blog would pay off some day.