No. The dealer needs to verify to their own satisfaction that an authorized person within the trust was the one they transferred the item to.
They may want to include a copy of a page or two with the name (s) associated with the trust with the 4473 as a cover their ass move but it realy is not needed as it is the end of the process and ATF already has the info on .. XYZ Trust and associated names.
The dealer realy only needs to verify at the time of transfer the person picking up the item is authorized. The transfer is already approved by way of a tax stamp and form.
No one gets a list of your accets. (Schedule A) including the ATF. Only the qualifying accet in schedule A that funded the trust as per your states rules.