I have one of those also, very similar but not quite identical to yours. F&W seems to have made the American ones of that exact type (the original was made by Webley of course), but IIRC other makers made them also.
There is now a book by George Layman, on the American "British Bulldogs", and I may get a copy. Check:
Layman calls it the "forgotten gun that really won the West", IMHO a pretty silly idea. (Can you imagine hundreds of settlers discarding their shotguns, Winchesters, and Colts and taking up Bulldogs to fend off an Indian attack? I can't.) Still, any information on those guns will be more than has been available. Surprisingly, there is a lot of variation, and I don't think I have seen two exactly alike.
As to the engraving on yours, it may look crude but it might be factory. An 1884 catalog from E.C. Meacham Arms Co., St. Louis, shows the F&W British Bulldog at $3.12, nickel plated, engraving $1.25 extra. Ivory handles were $1.25 extra, pearl $1.87 extra. Some modern collectors might consider the engraving and pearl grips in the same category as the famous lipstick, but at that time those guns were significantly better than many other guns that sold for less.
Still not an expensive gun; the Colt SAA was $13.75, engraving $2.50. Multiply by about $40 for today's dollars.