My experience signing with a sharpie, from my early turning years, is not good. Regardless of what it says on the package, sharpie is not permanent. It will fade (I've seen it turn sorta purple). It also reacts with solvents like alcohol, so most finishes will at least perturb it if not wash it away entirely. Better to go get a better quality pen (I sign with a burner now, so can't recommend anything - but the artist pens used by basket illusionists would be where I'd start).
I love sharpie in the shop because I can make depth marks on tools or draw lines on the bandsaw table - and it cleans up easily with a little denatured alcohol. But I won't use it for anything I want permanent.