Uh. Wow.
That looks like fun. Legality depends on how the law is written and interpreted.
From HRS, for convenience:
"Automatic firearm" means any firearm that shoots, is designed to shoot, or can be readily modified to shoot automatically more than one shot, without a manual reloading, by a single function of the trigger. This term shall also include the frame or receiver of any such firearm, any part designed and intended solely and exclusively, or any combination of parts designed and intended, for use in converting a firearm into an automatic firearm, and any combination of parts from which an automatic firearm can be assembled if the parts are in the possession or under the control of a single person."
Bolds added. The first defines automatic as a single function of the trigger... which a slidefire stock is not since it requires several actions other than the trigger to function. The second looks like it covers any assembly of parts that convert to an automatic weapon... but the slidefire has already been exempted from definition as an automatic weapon.
Looks like its legal. Makes me kinda want an AK.
But then... I really can't afford to feed such a thing. And it certainly would be fun arguing the finer points of the law with the nice officer inquiring about my "machine gun." The BATFE approval letter helps, though. I'd keep a copy of that sucker and the HRS definition of "Machine Gun" in the case with that puppy.