Three of the ideas so far I like:
1.Smaller capacitor-this is probably the simplest and cheapest option.
2.LED-this is also a good idea, the problem is that LEDs are (unless you want to start spending thousands of dollars for a pre-built option) hundreds of lumens, while most camera flashes are many thousands of lumens (I think). To get more than a few thousand lumens out of an LED source, especially when timing is critical, the only practical option is to make your own device, which will probably end up being rather complex and expensive. You would need a very high sensitivity camera and a very dark environment. CRI is not so much of an issue for most photography IMO, you can always artificially re-color stuff in photoshop.
3.Control discharge from the capacitor. If you can get a proper transistor or MOSFET or something (maybe a cheap MPSA42/MPSA92 will work?), you can put that in series with the flash bulb and have 100% control over the amount of time the bulb gets energy from the cap. Just something like
digitalWrite(flashPin, HIGH);
delayMicroseconds(flashTime);
digitalWrite(flashPin, LOW);