Generally it is legal to directly mail a damaged gun to an FFL BUT you live in New Jersey. I'll bet NJ considers a barrel to be a "complete firearm." In my state one would just mail the barrel in for a replacement or repair and it would be mailed right back to their doorstep.
A barreled action... yes, a firearm. Just a barrel, no. There must be a serialized part, just like any firearm... even if other states don’t consider it (airgun or blackpowder) a firearm.
In regards to shipping a long gun back, there is no issue in NJ... unless you are getting a replacement receiver (different serial number). And even then, there is no specific law that says it is illegal. If they have to send a completely different receiver, I would request them to send it to the same FFL... and do the transfer. Even though there isn’t a specific statue that says it is required, it is just safer to go that route than have a possibility of getting jammed up for it.
Since you went through an FFL, you have a FID. That allows you to possess/transport that muzzleloader without restrictions, like are present with handguns (to/from specific authorized locations, with no unreasonable deviations). You can box it up, go to USPS, and ship it back to them Priority. Coming back, it is legal for them to ship it back to you... as it is your firearm (no transfer of ownership). While NJ laws are confusing, there is no specific 2C statute that touches that process... minus the transportation laws (which are covered with you possessing a FID). If it is considered a firearm, treat it as such, and comply with Federal laws for shipping it back (no powder/primers in the box).
I did notice you were specific on them shipping back a replacement firearm... did they say that? If it is a barrel issue, I would think they would change the barrel. I would confirm that with the manufacturer... and even request to have the same receiver sent back to you. You can still send it to them yourself, but that would be the deciding factor on if it goes back to you directly or to your FFL. It may be worth it to talk to your FFL regarding it... but remember what I said about it being legal to ship it back yourself. If the FFL says you can’t do that (wants to make money on you to do it), then I’d personally find another FFL.
With that being said, I’ve shipped plenty of firearms via USPS, FedEx, and UPS. Never had an issue, nor has there ever been an issue with any L/E knowledge of doing so (including a background investigation with the NJSP, which I showed tracking information for a shotgun I sold during the application process). Even had a M1903, with a scope that I had mounted, left outside in the rain by USPS... which had Adult Signature required listed on the box. While I was pissed off about a soggy box, and possibly my rifle being wet (was fine, since my girlfriend came home early enough to bring it in, and smart enough to take it out of the box), leaving a firearm unattended for hours was more of an issue for me. Post Office spoke to the driver about it, who apologized for the incident.