You didn't ask for opinions on whether it should be refinished or not, so I won't offer one.
I believe the traditional inlay fill was engravers' wax.
I've never used it personally, so you'll have to do an internet search to find out exactly how to apply it. It's basically a rub on--rub off thing.
On my own engraving, I have had good luck with gloss black Krylon spray enamel. I spray a little into a paper cup, dip a thin, pointed wooden skewer into the paint and dab in onto the engraving lines. When nearly dry but still semi-soft ( a few minutes later) I clean the excess off the surface with 0000 steel wool.
This will be tricky to do, however, if working around a chipped finish. But with care, it could be done without damaging the original finish.
Glen Carson, on this forum, is a master engraver and may weigh in on this topic. Any suggestions he offers will be good as gold.