IANAL, nor 2A case law buff...but given the Troubles of 7 February 1991 ( https://en.wikipedia.org/wiki/Downing_Street_mortar_attack ) - and much other mischief that they'd be "just the thing" for - I suspect that U.S. law enforcement could be very interested in private owners of mortars.

That's partly why they set it up that way in the first place - ATF gets to review all those applications, and while they cannot reject them except for specific enumerated reasons, they can certainly investigate and/or notify other agencies.

Nevertheless, if you follow the procedure, it is legal, and rich collectors do it. Also applies to larger artillery pieces, tanks with working main gun etc.