If your client is a collector more than a shooter, I suspect that re-cutting anything would lower the value unless done by the factory. Perhaps that isn't a concern though.
Having a number of British guns some with worn "barrel" engraving, typically the rib, I am planning to eventually do this myself as well. I would note that if you haven't done so already take a very close look at the style of cuts used in the lettering. I find that at first glance what seems the same "font" between gun makers is cut subtly differently. I'd recommend planning the cuts on paper first . I have noticed that more refined gun descriptions at auction will note such things as refreshed engraving. I suspect it has a minor impact on value if done well. I wouldn't make it like new or deeper on an otherwise worn gun; only enough to improve legibility.
Jeremy
One more note on refreshing old gun lettering: duplicating the geometry of the original graver will likely be tricky so I'd experiment until you get a similar looking cut.
Jeremy