For Form 1 items you have to engrave your name (or trust name), city, and state, as you are 'making' a SBR 'somewhere' on the receiver where. The S/N needs to be the same (I believe), but can be re-engraved elsewhere as well, since your modification would destroy the original engraving.
Call the BATFE and get clarification. No, they are not going to come and shoot your dog.