IANAL and not as current on this as I was in the past:

Check for local legal issues both en-route and at the destination. You don't want to have to overnight outside of EWR (for example) if you were supposed to be connecting there, and the ongoing flight goes mechanical for 24 hours. This would apply to any thing considered be a firearm: ordinary or something you need a special BATF tax stamp for.

En-route or destination, it may be that local law requires a permit in addition to the federal permit, makes it outright illegal, or just refuses to issue any permits.

As far as I knew, the suppressor itself is legally a firearm, so just treat it the same. I'd probably put it in the rifle case too, inside a sock to protect the finish.