unless that's just personal preference, there are no legal issues with that... as long as its not a COMPLETE Class III firearm.
This is incorrect information. Actually, whether it's a Title II firearm or Title I firearm isn't the important distinction on whether you can leave the firearm with the engraver.
Engraving is gunsmithing according to the ATF, and requires that the engraver be an FFL.
http://www.atf.gov/firearms/faq/gunsmiths.html
If they do it while you wait, they don't have to enter it into their bound book. They don't have to make you fill out a 4473 to return it, either, even if you leave it with them.
Interestingly, they can repair NFA firearms as well without having an SOT, it appears. And the ATF "suggests" that you fill out a Form 5 for the transfer, but doesn't require the entire form filled out--just the face of it. Strange.
I don't know if there's a gray area where a non-FFL engraver can engrave your firearm as long as you are present, but that doesn't seem to an exception in the ATF's eyes.
If you want to use someplace local rather than one of the well-known places like Orion Arms, which you'd have to mail your receiver to, then I'd start asking local gunshops if they know any gunsmiths that do engraving.
Or do it yourself.
Aaron