Here is what I know to be fact and up-to-date LOCALLY
Gentlemen, I am NOT a lawyer, or a BATFE employee, or a 01 FFL.
I am a very frequent buyer and seller here on this Forum of handguns, predominantly S&W, but have sold Colt, Ruger and WWII Luger also.
I do hold a valid 03 FFL and have for years. 03 is Curios and Relics which the BATF defines as 50 years old minimum. As of today this means 1973 manufacture date, but we are less than 1/2 way thru 2023 so I personally use 1972 as a cut-off year, thus my most recent sale of a Model 25-2 with a ship date (per Letter) of February 14, 1972.
Now, how do I ship?
I do NOT use Bud's Ship my Gun because they only use UPS and no third party, kiosk or franchise UPS pickup point will accept. Only the Regional Customer Service Center which as posted by others, has NO PUBLIC ACCESS at any time.
I CAN use local 01 FFL's (LGS and Pawn Shops), but I DO NOT because they are extremely cost prohibitive! The lowest rate I have found is $75.00 plus shipping ($50) plus full value insurance ($50 and up). So a gun selling for $1000 adds $175 and they do NOT want it wrapped up or boxed because they have to enter the serial number onto their FFL logs. Most all are stores selling new and used guns and my transfer is nothing but competition to them, not really "extra money".
Because I personally want to wipe down the gun, place it into a gun rug (in-expensvive Allen brand), pack & protect, pad the entire shipment, including a plastic page sleeve containing the signed Bill-of-Sale, owners Manual, other docs, Factory Letter if I have them, a copy of my VA DL and my 03 FFL. This way I know (and photo) the shipment is clean, well packed and sealed.
I then use exclusively Federal Express!. Their Medium and Large, Overnight boxes are sturdy, and easily take all (the gun in the gun rug, bubble wrapped and page sleeve with documents). I double tape all corners to completely seal out all outside weather effects.
More than 2 years ago I had no trouble just presenting at Fed-Ex counter and they would print the label and away it went, however then (also as posted above) the CARRIERS changed their internal rules (NOT the BATF). They would no longer print the label IF you had to have ASR (Adult Signature Required). Please do not flame me for this statement as it may only apply to me and my local Fed-Ex station, so I don't know about you or your local Fed-Ex.
I was told that I have to print my own ASR label from home on my computer/printer and in order to do that I had to start a Fed-Ex Account which I did.
I was told by Fed-Ex employee that the reason is that with this method, YOU are the shipper of WHATEVER is in that package, they (Fed-Ex) doesn't know and doesn't want to know. Anything illegal is now YOUR problem...not theirs.
It is all fine with me as I get on my account click the send to name & address (usually a 01 FFL that I have already received from the buyers FFL a copy of their current and valid 01 License), click the ASR required, click the additional insurance (std included is only $100) and put in full value of my sale, then BEFORE I click print I can see the entire charges for the Overnight freight rate, the insurance and the charge for ASR (either a storefront, or residence) all totalled. If I'm OK with the total, hit print and out comes the label.
Put the label in the closeable/sealable pouch peel and stick the pouch onto the box and take it down to Fed-Ex station and tender it at the counter. Agent takes it and checks that my weight is the same on their scale as I weighed at home and they print me a acceptance receipt for the package. No money changes hands, no information is asked for or given. In..Out.. over and done.
The Tracking number is already printed on the label when I did it at home, so all is good to go. I get home I e-mail the Buyer the tracking number and send some pics of the shipment, and everybody's happy.
Again........The above works for me, here in Central Atlantic Viginia.........I don't know what may work for you, but I've sold and shipped handguns, long guns, and even ammo (just have to click Fed_ex Ground...can't go air or next day). Haven't had any problems and I believe the Buyers have been satisfied too.
If you need any more details just PM me and I'll do my best to assist, but again...I'm not a Lawyer, or employee of Fed-Ex or BATFE.
PS I have already included my estimate of costs when I first advertise the gun for sale, because I have found (especially with SWCA For Sale Forum) that one price noted as "shipped and insured" is better than saying buyer pays shipment and insurance later at actual cost.
If my shipped and insured gun price is too high, the Forum will let you know by letting your add sit for weeks until you get the idea and reduce.

So far I haven't had anyone tell me price is too low!