What I've ended up doing is setting a task to watch for the backup complete event in the Application log.
Steps to replicate this if anyone is interested are:
- Open Task Scheduler
- Create a new basic task
- Enter a description for the task
- On the trigger screen select "When a specific event is logged"
- Select the Application from the log drop down menu.
- Select Backup from the source drop down menu.
- Enter 4098 as the event ID..
- On the action screen select "Start a program"
- Enter "Shutdown" into the Program/Script text box.
- Enter "/s" into the Arguments box.
- Click next & Finish.
- Run a backup to confirm the task is working.