CFR 178.147 addresses repair/replacement of firearms:
https://www.govinfo.gov/content/pkg/CFR-2002-title27-vol1/pdf/CFR-2002-title27-vol1-sec178-147.pdf
The explanation you were provided, is incorrect.
The replacement must be of "same kind and type"- would mean no change in chambering. Makes sense, as caliber is recorded in the bound book along with the manufacturer info.
Makes no difference whether you send the firearm to a licensee, or an FFL sends it... and it's entirely illegal to "reissue" an identical S/N on another rifle. The defective firearm- with its unique S/N would be recorded as destroyed. That S/N cannot be "reused".
Typically they would destroy the receiver if it is not serviceable, and send you a replacement firearm with a different S/N.