I'm not a HID expert, but I did install one in my KLR this summer.
What happened to me was the ballast didn't like the draw when I hit the starter. If I didn't let the bulb come up to full temp before I started the bike, the light would go out. The company said that the ballast didn't like "hot re-strikes" and that the ballast was designed to do that to protect it. The only thing that I can suggest is that the current to the ballast is dropping and causing it to shut off...odd when there should be plenty of power to it. Have you wired it (through a relay) to the battery?
Sorry, that's all the help I can give.