Short answer:
You can transport your SBR interstate without approval if you do the following: Separate the upper and lower of your SBR, transport them separately, and once you cross state lines don't let them get near each other until you have the approved Form 5320.20 in hand.
Long answer (with citations):
This same thing happened to a customer of mine: His Form 4s for both his SBRs came back just a week before he was making an interstate move. Our store was in the middle of an ATF inspection at the time, and my customer wanted me to ask the lead ATF agent if he could expedite his Form 5320.20 (interstate transfer form). The ATF agent said he couldn't, but he added that if the SBRs were disassembled into uppers and lowers and transported separately it would be perfectly legal. He said that as long as the SBR uppers and lowers weren't re-assembled (or transported or stored too close to each other) before the approved Form 5320.20 came back, no laws or regulations would be broken.
But you don't need to take my word for it. Here is an excerpt from
section 2.5 of the the ATF's NFA handbook where they point out that an SBR isn't considered an SBR unless it's currently in SBR configuration:
the ATF said:
Firearms, except machineguns and silencers, that are subject to the NFA fall within the various definitions due to specific features. If the particular feature that causes a firearm to be regulated by the NFA is eliminated or modified, the resulting weapon is no longer an NFA weapon.
For example, a shotgun with a barrel length of 15 inches is an NFA weapon. If the 15- inch barrel is removed and disposed of, the remaining firearm is not subject to the NFA because it has no barrel. Likewise, if the 15 inch barrel is modified by permanently attaching an extension such that the barrel length is at least 18 inches and the overall length of the weapon is at least 26 inches, the modified firearm is not subject to the NFA.
So that means if you separate the upper and lower of an SBR, you aren't in possession of an SBR and you can transport the parts interstate just like it was a regular firearm. Don't forget that once you get across state lines, not only can you not re-assemble the upper and lower until your Form 5320.20 comes back, but you also can't have the upper and lower in "close proximity" to each other with no other "useful purpose" for having them.
While it's true that an SBR isn't an SBR unless it's assembled, there are situations where by merely having the parts near each other you could be considered to have the intent to assemble them. This situation is outlined in
ATF Ruling 2011-4. Basically, if you have no other "useful purpose" for having those parts other than to illegally assemble an SBR, those parts must also be in "close proximity" to each other to be illegal.
What's the exact definition of "close proximity"? I have no clue. I asked that ATF agent and he told me to ask a lawyer. I recently moved interstate and the same thing happened to me: My Form 5320.20 didn't come back before I moved, so I put my SBR lower in one car and my two short uppers in another car and I made sure there were no other lowers in the car with the uppers. When I got to my new house, it took a week for my form to come back, during that time I just left my short uppers locked in my car. I'm not a lawyer, but it seemed to me that wasn't "close proximity", so I took my chances.
Obviously, this is only regarding federal laws and regulations. It's always possible you could have issues with local or state laws.