Nice gun that carries some value.
This is most certainly a job for a barrel expert, with experience with British guns.
The critical wall thickness measurement is the end of the chamber and at the forcing cones. Heavy walled U.S. 12g doubles usually have greater wall thickness at the forcing cone because the angle of the cone exceeds that of the external taper of the barrel - not so light weight 12g British and small bore guns.

We are fortunate to have several experts, and because they are experts you may have a wait.
I don't mind listing them, and pray I don't forget someone.

Kirk Merrington
Kerrville, TX
http://www.kirkmerrington.net/
830-367-2937
Keith Hackleman (apprentice)
303-919-7376 cell

Dean Harris
Skeets Gun Shop
Tahlequah, OK
918-456-4749

Champlin Arms
Enid, OK
http://www.champlinarms.com/Default.aspx?tabid=28

Steve Bertram
Bertram & Co
http://www.bertramandco.com/
Boulder CO
303-938-1996

Brad Bachelder
700 Plymouth N.E.
Grand Rapids, MI 49505
616-459-3636
Lori@mastergunmakers.com

Hugh Lomas
920-876-3745
Elkhart Lake, WI.

Dennis Potter
Muskego, WI
414-425-4830
dennispotter@att.net

Not to be dramatic, but done wrong looks like this



You might consider forcing cone lengthening only, which does not take a gun out of proof (this should start a debate wink
More thoughts here
https://docs.google.com/document/d/1ZIo0y746UsSRZIgRuuxwAbZjSBHitO_EanvwLYc-kGA/edit