Note that it requires a special FFL to own a grenade launcher, ...
I'm not so sure about that, most grenade launchers are fairly short, the M79 has a 14 inch barrel, the M203 so the restriction is more likely,
Under the National Firearms Act (NFA), it is illegal for a private citizen to possess a sawed-off modern smokeless powder shotgun (a shotgun with a barrel length shorter than 18 inches (46 cm) or an overall length shorter than 26 inches (66 cm)) (under U.S.C. Title II), without a tax-paid registration from the Bureau of Alcohol, Tobacco, Firearms and Explosives, requiring a background check and either a $200 or $5 tax for every transfer, depending upon the specific manufacturing circumstances of the particular sawed-off modern shotgun being transferred. Sawed-off shotgun
that they are sawed-off shotguns, rather than because they are grenade launchers, the issue of the grenades are a separate issue, non-explosive projectiles should be ok. If the grenade launcher were long enough it would be considered a 1 1/4 guage shotgun.
Of course IANAL so consult a trusted advisor for anything other than hypothetical banter.