I've done some more reading on this, found this very useful article: https://ricksfreeautorepairadvice.com/troubleshoot-hid-headlights-problems/
So it sounds like it's the bulb - the ballast eventually overheats from trying to keep restarting the bulb and gives up - but turning off and on the power to the lights will 'reset' the ballast and it will keep driving it for a while longer. Thankfully this should be an easy and cheap fix..