Hi
If you load it with a solid ball it is not a shotgun. You need it on your FAC.
I took advice on exactly this scenario when renewing an FAC holders ticket several years ago.
I'm retired now but as far as I am aware the law has not changed

Re the parlour pistol. You say its modern manufacture so no Section 58 exemption then. (Modern is held to be manufactured after 1939).
In the first instance it is definately not a Section 2 firearm as it will have a barrel less than 24 inches.
Is it capable of inflicting a lethal injury? (This held to be in excess of 1 joule). If yes then it is a lethal barreled weapon and therefore a firearm.
It is most certainly not an airgun as the means of propulsion is the chemical reaction/gas effect of an exploding percussion cap and not compressed air /CO 2 etc.
It is therefore a section 1 muzzle loading pistol and must be entered on your FAC if you want to shoot it!