Lots of bad advice here. Yes you can ship it. No you can't ship to your brother legally.
The ship to yourself rule allows you to ship your firearm to yourself for your own use, the firearm not being transferred into the possession of another. Say I'm planning a SD pheasant hunt. I can lawfully ship my shotgun to myself in care of the outfit I booked with. Typically the package is held for you unopened so a "transfer" cannot be claimed.
The prohibition is the transfer of a firearm crossing state lines to a non-licensee, transfer being physical possession So technically leaving the firearm with you constituted an illegal transfer. Not that any agent in his right mind would waste his time with such a trivial and innocent matter. But don't compound the matter bu using a subterfuge to effect a transfer back to him.
Per firearms law anybody can ship to a license holder. Easiest thing is to have your FFL ship to his FFL and for a few bucks extra not have to worry about it.
This will all be somewhere on the ATF site. Probably you can suss it out from the Q&A page. Good luck with the Post Office or common carrier's site. Last time the postmaster said no, I handed her their own regulations, she called somewhere and eventually said yes. Much easier to use your FFL who has already been through all that.
This^^^ Why take the chance, pay the transfer fee. ship to a FFL.
From the ATF Q&A Page: I was looking specifically st the Post Office. Fed Ex and UPS seem to have their own rules which are on a fly wheel and it just depends on where that wheel stops or who you ask. They don't know their own policy. Once told me they couldn't ship a scope because it was part of a firearm!!!
May a nonlicensee ship a firearm through the U.S. Postal Service?
A nonlicensee may not transfer a firearm to a non-licensed resident of another state. A nonlicensee may mail a shotgun or rifle to a resident of their own state or to a licensee in any state.
The U.S. Postal Service recommends that long guns be sent by registered mail and that no marking of any kind which would indicate the nature of the contents be placed on the outside of any parcel containing firearms.
Handguns are not mailable. A common or contract carrier must be used to ship a handgun.
[18 U.S.C. 1715, 922(a)(5) and 922 (a)(2)(A); 27 CFR 478.31]
