You can absolutely send it through USPS.

https://pe.usps.com/text/pub52/pub52c4_009.htm
Section 432.3 subsection d.

Rifles and shotguns may be mailed by a non-FFL owner domestically to a FFL dealer, manufacturer, or importer in any state. These items must be mailed using a class of mail, product, or Extra Service that provides tracking and signature capture at delivery.