While I'm not making any statement about the feasibility of the Skarp, I think you might be exaggerating the size of a moderately powerful modern lasers. At my company's R&D lab I have some 1W CO2 lasers that, similar to what what you describe, are about the size of loaf of bread. They run on three phase power, have inch-thick cables and required hilariously large cooling shrouds. Two facts about those types of lasers: They are cavity lasers designed to have very good beam quality, and they are OLD.

I have a cabinet full of other types of newer, moderately powerful laser modules which vary from pin diodes to modules the size of a AA battery, to things the size of cigars. Some of them are pretty darned powerful. Any laser modules that are water cooled would DEFINITELY not be something you would want to hold up to your face, but I also don't think that's the kind of power required for cutting hair. I'm not saying that any laser running on a AAA battery at a useful power could last a useful amount of time, if you could figure out some secret sauce method for very quickly coupling a high intensity pulse (hairs have a very small area, after all) onto some hairs at a wavelength that they like to soak up then it seems.... possible. In theory the laser wavelength and power could even be tuned dynamically to match the hairs, that is if they had somehow figured out a way to fit a Mach-Zender modulator, an electro-optical modulator and a frequency comb into a razor handle--ha!

Anyhow, I might be wrong in all this, but my point is mostly that it wouldn't take a water cooled laser to burn some hair. Come visit me in Vancouver and we can test it out (on your beard).