Legal:
The Bureau of Alcohol, Tobacco, Firearms, and Explosives, referred to here as the ATF, is the government agency that writes regulations to control things like your Bowling Ball Mortar. Fortunately for us and for amateur artillerists everywhere, there is a specific exemption in the ATFs rulings that allows us to build your Bowling Ball Mortar without fear of legal troubles. According to their rule, which is posted below, your Bowling Ball Mortar is considered to be a replica, and is therefore not something that carries any special rules or regulations on a federal level. Below is the relevant excerpt from federal laws that fairly clearly (comparatively speaking) explains that your Bowling Ball Mortar is in fact a benign replica:
26 USC sec. 5845(g) "Antique firearm.-The term 'antique
firearm' means any firearm not designed or redesigned for
using rim fire or conventional center fire ignition with
fixed ammunition and manufactured in or before 1898
(including any matchlock, flintlock, percussion cap, or
similar type of ignition system or replicas thereof, whether
actually manufactured before or after the year 1898) and
also any firearm using fixed ammunition manufactured in or
before 1898, for which ammunition is no longer manufactured
in the United States and is not readily available in the
ordinary channels of commercial trade."
Because your Bowling Ball Mortar is loaded with the propellant separate from the projectile, uses simple cannon fuse for ignition, and is patterned off of a Seacoast Mortar in use in the 1860s, it is falls clearly under the federal definition of a replica firearm.
Even though your Bowling Ball Mortar falls under the federal definition of a replica firearm, the laws of your state, county, or municipality may be different than federal law and have more stringent rules. It is up to YOU to know what laws dictate the construction, possession, and operation of your Bowling Ball Mortar in your area.