A second test confirmed that this is consistent behavior unfortunately. I think what is happening is that the turn on current draw of the steppers is too high for the computer PSU and it trips the overload protection. To test this what I'm going to do is connect my bench power supply in parallel with the computer supply and then after starting the motors wind down the current. That should tell me if the computer PSU will handle the base load current.
If it will then I will probably put some kind of large capacitor across the 12V rail and GND to help minimize the surge. I may do a quick writeup on the display arrangement I finally got working, particularly if there is some interest there.
I'll report back how I go with this.