Gun engravers and gunsmiths are not the same breed of cats. Good gun engravers are a rare species and normally have work sent to them. Factory engravers work mostly or sometimes exclusively for one company. In any case, an engraver is primarily an artist, not a gun repairer.
I suggest you do a Google search (as I did) on "gun engraver" and check on the location of engravers. Then contact several, describe what you want done and request an estimate, both of money and time. Also ask for pictures of their work; some very fine engravers do work that is not to everyone's taste, so seeing examples can help you decide. (Most gun engravers have FFLs, shipping to them is not a problem.)
It depends on what you want, but I suggest avoiding local jewelry and trophy engravers. Few have any experience working in steel, and their work is often disappointing.
Another point is that some guns are quite hard and engravers won't/can't work on them. Factories generally have engraving done before parts are hardened, avoiding that problem.
HTH
Jim