Hi John,
This answer is too late for you but may help others. Yes, you can replace the cable. You will need a 75mm Security Tamper proof snake tongue screwdriver bitsize 4mm, which you can get off t'internet. Breville and other irons have supposedly tamper-proof screws, but the above works. Then its a straightforward live/neutral/earth rewire inside the iron. Could not be much easier.
Tom O
August 2019