NO !! Despite journalistic inaccuracies, this weapon does not *FIRE* ! You can "pull" (as on the trigger lever) or "shoot" (as in ordering some minion to do so) but not "fire" it !! Therefore, it does not need a FIREarms permit !!
However, you may need a local authority permit to put in in your garden as it constitutes a "change of the local scenery" !! You will also need a large tent or tarp or else it will be painted in urban camouflage by the local avian lifeforms !!
I hope this is helpful !!
@If they only got 100 yards range they made it wrong. - The Mongols used glue boiled out of the hooves of their animals (cattle, sheep or horses). It has the remarkable property of being slightly elastic while remaining fairly rigid !! Being organic in nature, it also binds the horn pieces to the wood without too much differences in thermal expansion coefficients. Otherwise, it would have been a tad embarrassing for the Mongol to try and shoot that bow in summer only to find it falling apart in his hands because he made it in winter (when he's resting from his wars) !!
BTW, Mongol bows are known to be effective at 400 yards and will drive through plate armour at 200 yards; something the Byzantine cataphracts discovered to their great dismay. The Mongols also went into battle with a string of ponies, irritated the cataphracts with their archery, forced the heavy cataphracts to chase them, jumped from pony to pony to tire out the Byzantines and then slaughtered them when they were helpless. Not fair, but war is war !!