Ash,
If you have a Scan Guage, set it to display the voltage. I believe that the Scan Guage will react quicker to voltage changes than any cheapo multimeter that you would use. The only issue with that though is that you won't be able to read the voltage changes during engine start. You said that you didn't turn the lights on until the VX was already running anyway.
It sounds to me as if the lights are a poor design. They should have be manufactured with pull down resistors already included.
Vehicle power is notoriously 'dirty'. Even with the voltage regulator. 24V Vehicle Power Conditioners (VPC) are common on modern military vehicles due to the amount of electronics they now run. I don't think that there are commercial 12V VPCs available though.
You can do that test that they mentioned without removing the lights from the vehicle. Just disconnect the power lead and connect it to a wire directly from the battery.
Another note: since LEDs do react much quicker to voltage changes than incandecent bulbs, it's a good idea to always solder all of the connections rather than just pigtailing & taping them.