The maker’s / sponsor’s mark is definitely that of Edward Barnard & Sons and was used on silver assayed in London from 1829 until 1846. However, after some further research, I believe that your coffee pot is silver plate, perhaps what is called Old Sheffield Plate (OSP) rather than electroplate. Barnards used a shield-with-cross trademark (derived from the arms of the city of London) on their silver plate and I suspect that the A may be some sort of date coding but I don’t think that there is any record of these marks.
OSP is produced by fusing a silver sheet onto a copper base and therefore contains substantially more silver than electroplate which comprises an extremely thin layer of pure silver deposited onto a base metal by an electrochemical process. If your coffee pot is OSP you may be able to detect the 2 layers by looking closely at its edges, on the lid for example.
You may also find the comments on this post on another forum interesting.