That is not the W.& C. Scott that is well known. William Scott was a late apprentice - most apprenticeships began at age 14, but he began at age 21. He finished his apprenticeship in 1834 and became a "gun finisher" for other companies. In 1838 his brother Charles joined him in the business and formed W. & C. Scott. There was never a "Scott and Son's" related to this maker. In 1858 the business became W. & C. Scott and sons. With this information in mind, we are looking for a different "Scott". Interestingly, in 1871 W. & C. Scott and Sons used a "Tower with a Flag" stamp on their guns. Your barrels seem to have a tower stamp of sorts, but the time period and name on the locks does not match the authentic utilization of that.
Unless the photo is poor, the London Fine Twist appears stamped, and is not the right font, nor consistency for a "best quality" gun.
The Proof mark with B-P-C is from Birmingham Proof House and was 1868-1925. The crown with BP is from 1856 and is the Birmingham Provincial Proof for barrels.
In 1870, 11 gauge was tapering off and was actually made in breach-loaders with cartridges.
The Book of Provincial Gunmakers up to 1850 does not list a "Scott".
The Provincial proof marks make sense because they were for guns by a person or company that was not enrolled in the London Gunmakers Company.
A search of Indemnities and Applications to the "Company" shows who had an apprenticeship and, upon completion, who applied for freedom in the "Company" to be a registered gunmaker. I can only find William Scott's son's with either or both. There is a J. Scott that applied for freedom in 1849, but he has no sons or apprentices and was not independently in business for long.
The HL could be Herbert Charles Lambert, who was an exporter of silver and sporting goods, but he was more prominent in late 1800's and even early 1900's. There was HL prefix used by Halloway and Co. London on barrels they made for trade guns, but not their own.
Your gun has no "vents", which are commonly platinum. It also features cleft hammers. These were common on trade or guild guns. Most of the registered Makers had some uniqueness to their hammers, whether in shape, engraving or finishing.
There were a lot of English and Continental guns being imported at the conclusion of the civil war, as the American makers were tapped with supplying arms and military items.
Based on all of the above, I believe you actually have a later trade gun for export, made between 1865-1870. I hope this is helpful.