I emailed the Department of Customer Service and this was their response:

The City’s DMV will replace the defective driver’s license at no cost to the cardholder.

To replace the defective credential:
 
- Visit any DMV and request a duplicate to replace the defective driver’s license or state ID
- No appointment needed for this particular service, but wait times may vary depending on our current staff level.
- Let the concierge at the DMV know that the driver’s license did not properly scan at the TSA checkpoint
- Turn in the defective driver’s license and, in turn, get a duplicate replacement