unless the receiving FFL demands it be sent to him by a FFL in your state ( and some FFLs do require this type of transfer). it's just way simpler and much more easier for you to ship the firearm to the FFL.
in my opinion, it's simply not worth paying the extra money (fee) to have it shipped by a FFL. the ATF doesn't require it, so why jump that hoop when you don't need to?
the only due diligent on your part, is to check his 01FFL number on the ATF EZ check ( you don't need a physical copy of his license, thought it would be nice to have), and mail it to the business address that is on the EZ CHECK. then just screen capture that webpage. print it out. that along with the copy of the shipping invoice is proof enough you followed the law.
in all the times I shipped a firearm, I shipped it myself and added the shipping and insurance to the price of the firearm ie. SHIPPING AND INSURANCE INCLUDED. that way the person who is buying the gun knows up front the total cost.
now, some folks are so cheap that they may decline to pay the few bucks extra for insurance. which I have not that happened to me yet. but if they did, I would just deduct the insurance cost from the total, and write back, when they send me the money to include a handwritten letter declining the insurance and have their signature notarized. that will cover me, if the gun got lost, damaged etc.... .