Just in case anyone has the same issue in future: the display is a Sharp LQ9D340H. They are available on eBay for around $150 new, so I've ordered one as a replacement.
Getting to them is easy: there are a number of countersunk head screws that secure the front panel to the main frame. To remove the ones at the top you need to remove the trim strip. After undoing them all, the front panel can be gently pulled forward until it clears the inputs. The two screws that secure the display cover can then be removed, and the cover levered up and removed. Then undo the 4 screws that secure the LCD display. It has a 2 pin connector for the backlight and another connector for the drive signals - remove both and then fit to the new LCD and re-assemble.