Hi Lucy and Stephen,
At the moment, I don't believe it's possible to have a badge 'expire' alongside a certification or audience membership. However I can see why this would be useful!
I had a look in our product feedback tool and it looks like this idea has a few votes already, so I'd suggest that you also add your vote to this to help our Product team understand it's priority.
To do so, go to our Suggest features dashboard and select Make a suggestion then Feedback. If you search for something like 'expire badge' there you'll see the item 'Badge criteria to be linked to certifications…' where you can add your vote.