Beech:
While rebarreling is certainly a possibility, I looked at your gun on the Hill Rod and Gun website and it does not look like either my boxlock or the A. Lancaster hammergun pictured here on the for sale board by Turnagain. Both his hammergun and my boxlock have A. Lancaster/ London on a scrolled banner on the side of the action, very similar if not identical engraving, and palm fronds engraved on the fences.
I am afraid I agree with Rocketman that you may not have an A. Lancaster. On the other hand, I am not sure that it matters a whit if you have a good functional bird gun that you paid a reasonable sum for. Just enjoy it!
Dr. P