I'm going to do a little bit of copying and pasting from Wikipedia which as we all know is the fount of all knowledge:
In practice, the CW has a number of drawbacks. As the number of stages is increased, the voltages of the higher stages begin to "sag", primarily due to the electrical impedance of the capacitors in the lower stages. And, when supplying an output current, the voltage ripple rapidly increases as the number of stages is increased. For these reasons, CW multipliers with large number of stages are used only where relatively low output current is required.
So yes, you can expect it to not provide much current, and maybe the capacitors are trying to draw too much which is making the output sag too much (and the ripple become a real problem). The article goes on to say:
These effects can be partially compensated by increasing the capacitance in the lower stages, by increasing the frequency of the input power and by using an AC power source with a square or triangular shaped waveform.
Another thing it says is:
The biggest advantage of such circuits is that the voltage across each stage of the cascade is equal to only twice the peak input voltage in a half wave rectifier. In a full wave rectifier it is three times the input voltage.
So switching to a full-wave design instead of a half-wave design would reduce the number of stages for the same output voltage, and thus should also allow an increased current.
(Wikipedia, we worship you :) )
If the flash lamp will handle the power, your unit should do the job. However, you need a much bigger capacitor.
As you've calculated, 100 joules in 350 usec is 285 kW, assuming constant current and voltage. At 1 kV, this is a current of 285 amps. the nbu max is 1000 amps for pulses less than 1 msec, so you're fine there.
However, if you've sized the cap so that its energy is 100 joules at 1000 v, it will be completely discharged at the end of the pulse, and so the current cannot be constant.
Let's say you want the capacitor voltage to be 1000 volts at the beginning of the pulse, and 900 volts at the end. That says that your capacitance needs to be about 1000 uF.
Also, as you know you need to find the flash equivalent resistance. In order to get 285 amps at 1000 v, you need an equivalent resistance of less than 1000 / 285 ohms, or 3.5 ohms. Allowing for stray resistances in the wiring and the nbu, you'll need a maximum flash resistance on the order of 2.5 ohms. Any more and you'll need a higher voltage on the cap, and the nbu won't work.
The volt rating on a capacitor tells you at what voltage a capacitor breaks down. When it breaks down the capacitor will burn through its dielectric causing a short. There is nothing wrong with using a capacitor with a higher volt tolerance, just make sure the capacitor has the same capacitance and that both are polar or non-polar capacitors.