I know the procedure for the 01 FFL is walking in and showing a license and telling them its a handgun.
There's supposed to be other paperwork required, and when I've shipped a handgun through an FFL (but took it to the post office myself FOR the FFL), there was an additional form required. (But not because it was a handgun.) The regs specify the form. I don't know how rigorous the FFLs and Post Office employees are about that extra paperwork. I suspect some are strict and others less so.
(For years, one of the FFLs I used was with short walking distance of the local post office -- so shipping by mail was easy. I shipped a number of guns through him, but acted as his agent by carrying the guns to the post office for him. He was the transfer agent. HE addressed the package,etc., etc., and it was recorded in his "bound book."
IF someone ships it to you, just accept it and smile -- somebody saved a lot of dollars. If there's a problem, it's their problem. Just be wary of shipping a handgun yourself through the US Postal Service. I really wish 03s could do it -- and it's counter-intuitive that we can't.
I had investigated 03 usage some time back, but more recently found other issues I thought I understood, but did not; that was cleared up in a discussion like this, on this forum. (My misunderstanding wouldn't have led to legal problems, but it a pretty bad misinterpretation/misunderstanding.)
The regs are hard to interpret, and sometimes easy to misinterpret. I keep learning how much I don't understand.