Re: Henry Beckwith Pinfire 12 ga.
You haven't shown or listed the actual markings; is it marked Wm Beckwith and does it have Birmingham proofs? I ask because British pinfires are far less common than those made in Belgium.
Boothroyd's Directory of British Gunmakers lists William Andrew Beckwith 1808-1865, in business in Birmingham 1810-1861, and business owned by William and Henry Beckwith 1865.
I can't suggest a value, but pinfires were never popular in USA, and are not highly sought by US collectors today.