If you want another option for really cheap HV power supply you could look at CCFL inverters. You can get ones designed for CCFL headlight rings designed to work off 12v. That say 5-8ma output at 900v and really cheap at $3us with shipping. You could use it as in input to a common transformer and step it down. A standard 110v > 48v would give you 900v > 360v 12.5-20ma
They also come in super tiny sizes for laptop screens that would be easy to fit inside a timing gun.
My guess would be yes it does. I think this is why they use Xenon bulbs as the metal around the bulb is used to trigger the existing charge to emit light and is done without any circuity adding hysteresis. The charge from the plug wire pushes the voltage in the tube above its ignition voltage and viola a burst of light. At least in the simpler schematic. In the fancier one you have a 1-4kv trigger transformer and added circuitry that would likely add some lag.
I wonder what kind of amperage you get at that voltage.