Check state and local laws. In some places, only FFLs are allowed to ship guns, or the FFLs are only allowed to receive guns shipped from other FFLs.
I'm not familiar with any FFLs that will complete a transfer without a copy of the ID or FFL of the sending party.
Best thing to do is contact the FFL before you ship and make sure you have all the things they require on their end in place. You might still be subject to laws at your end that they would have no way of knowing about.
Can you list a single state where only FFLs are allowed to ship guns?
Can you list a single state where FFLs may only receive from another FFL?
CA has a goofy deal where FFLs wishing to ship to CA have to register with them... yet nonFFLs may ship to FFLs without registering.
As an FFL well into my 4th decade with an FFL I can assure you any FFL with any sense is perfectly happy to receive a firearm from an individual without a copy of their ID. Their ID would be valueless as it is completely unverifiable by the FFL.
All the law requires is a name and address for the bound book. If the FFL maintains a copy of all the people sending guns he may very well be breaking the law by maintaining a list of potential gun owners outside the bound book.
The standard argument that they are covering themselves is absurd...