Lots of detail to be found here:
http://www.zr1specialist.com/HAT%20W...20Problems.pdf
If the voltmeter isn't dropping, it might be worth shorting the clutch safety switch (have a look at the last paragraph in the PDF above).
For what it's worth, I fought the "no start" intermittently for a while, including shining up the solenoid contacts. Finally bit the bullet and did the DNS Relay Mod - completely worth it for the peace of mind.
http://www.zr1.net/forum/showthread.php?p=88156