Are there any examples of using laser as a precise flash source for photography?

Yes. Laser light sources are used as "flash" of a duration down to a couple of femtoseconds. However the setup isn't one that I would refer to as a flash source for photography, nor can it be easily, nor cheaply, assembled on a typical hobbyist bench. Thorlabs for example offers off-the-shelf such femtosecond laser source.

Are there any intrinsic problems with this kind of setup?

• the speckle produced by the laser itself (also pointed out in a comment by Chris H), which looks like this and will produce a grainy appearance of your object:

Speckle can be reduced by the use of a "despeckler" (a device which vibrates in front of the laser beam and mitigates the speckle)

• controlling and triggering the laser to produce the brief flash of light you are looking for; a laser diode is similar to an LED in some ways, but it is far more sensitive to ESD, and can't be overdriven even for a short time (which many LED can, though it would reduce their life time). Your objects fly at 300m/s which is close to the speed of sound, this can be very fast depending on your imaging system's magnification ratio, hence, I suppose, your 100ns flash duration. Going into the electronic required for this may go beyond the scope of photo SE.
• producing an extended beam of light, of a size at least equal to your object. For this you will need a beam expander, this document from Newport shows you how to build one, you can buy one also from edmund optics or thorlabs.