Not legally.
An out-of-state FFL is not registered with NJ and does not have the forms required to legally execute the sale.
I think you are confusing requirements placed on NJ FFLs with requirements placed on NJ buyers. An FFL in NJ is required to submit paperwork to the Superintendent, but there is no requirement that an individual register his long gun. This is not a condition of sale, delivery or receipt: it is a condition of being an FFL holder in NJ. A dealer outside of NJ is not required to follow NJ laws.
These are the conditions of buying a long gun in NJ:
13:54-3.12 Requirements for sale of rifle, shotgun, antique cannon or receiver
Every purchaser, other than a licensed or registered retail or wholesale dealer or manufacturer, or law enforcement
agency as provided by this chapter, must present a valid firearms purchaser identification card to the dealer who is
required to confirm the identity of the purchaser. The purchaser must also sign a certificate of eligibility (NJSP 634),
filled out by the dealer or his licensed employee, for each rifle, shotgun, antique cannon or receiver purchased. The
certificate shall indicate that the purchaser presently complies with all of the requirements for obtaining an identification
card, and shall set forth the purchaser's name, address and identification card number. The certificate shall be retained
by the dealer. Prior to completing the sale or transfer of a rifle, shotgun, or receiver, the retail dealer shall initiate a NICS
check with the Division of State Police.
The OP can bring his FID and a certificate of eligibility to a Texas FFL holder, and the Texas dealer will place copies with the 4473. The dealer will also conduct a background check with NICS.