You're not transfering ownership and there's no such requirement (to the best of my knowledge) requiring an FFL for service or repairs of a firearm - and the 'action' even when stripped is considered a firearm.... and engraving is considered to be a service.
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"I'm a Setter man.
Not because I think they're better than the other breeds,
but because I'm a romantic - stuck on tradition - and to me, a Setter just "belongs" in the grouse picture."
George King, "That's Ruff", 2010 - a timeless classic.
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