I would be extremely careful.
The USPS requires that form 1508 be filed when certain firearms are shipped. On this form, you are certifying the following:
"The undersigned is a manufacturer of firearms or bona fide dealer therein, and the parcels presented for mailing herewith are customary trade shipments or other articles for repair or replacement of parts. To the best of my knowledge and belief, the addresses are manufacturers of firearms or bona fide dealers therein."
By BATFE definition, a C&R holder is not a "bona fide dealer"
The applicable provision in Title 18 is copied below....
FIREARMS LAW ADMINISTERED BY THE POSTAL SERVICE
TITLE 18, UNITED STATES CODE, CHAPTER 83
§ 1715. Firearms as nonmailable; regulations.
Pistols, revolvers, and other firearms capable of being concealed on the person are nonmailable and shall not be deposited in or carried by the mails or delivered by any officer or employee of the Postal Service. Such articles may be conveyed in the mails, under such regulations as the Postal Service shall prescribe, for use in connection with their official duty, to officers of the Army, Navy, Air Force, Coast Guard, Marine Corps, or Organized Reserve Corps; to officers of the National Guard or Militia of a State, Territory, Commonwealth, Possession, or District; to officers of the United States or of a State, Territory, Commonwealth, Possession, or District whose official duty is to serve warrants of arrest or commitments; to employees of the Postal Service;
to officers and employees of enforcement agencies of the United
States; and to watchmen engaged in guarding the property of the United
States, a State, Territory, Commonwealth, Possession, or District. Such
articles also may be conveyed in the mails to manufacturers of firearms or
bona fide dealers therein in customary trade shipments, including such articles for repairs or replacement of parts, from one to the other, under such regulations as the Postal Service shall prescribe. Whoever knowingly deposits for mailing or delivery, or knowingly causes to be delivered by mail according to the direction thereon, or at any place to which it is directed to be delivered by the person to whom it is addressed, any pistol, revolver, or firearm declared nonmailable by this section, shall be fined under this title or imprisoned not more than two years, or both.
The short answer to the OPs question is yes you can, but you have to use a common carrier (UPS, FedEx, etc.), so long as your State's laws don't prohibit it on either end.