Conor:
No need for unscheduled leave. Just do the print outs as you’re told to do by DVLA when your card is faulty. Keep them on you and as they meet the legal requirement for recording your work then there’s no issue. Hand them in after 28 days. It’s no different than working on agency when it used to be paper charts other than this is now a computer print out with human readable times instead of a squiggle scratched onto a paper disc.
At one of the places I work, a DHL contract, they will not allow agency drivers to start a shift without a fully-functioning digi card.