Do NOT carry the suppressor in your carry on bag - I don’t care what people tell you that they did it with no issues or that it isn’t a firearm in and of itself.
The TSA considers it a prohibited item for travel in carryon and can fine you 1k.
It goes in your checked baggage and you can verbally declare it to the ticket agent when you give them your luggage as an added measure of notification.
Lastly you can ship and insure it in advance of your trip to a trusted party where your traveling that way if it gets stolen you get some money back, in case you don’t trust putting it in your checked luggage.
Link to TSA fines below, you’ll see the fine amount here but no where else on TSA’s public facing systems :https://www.tsa.gov/travel/civil-enforcement