Like the other 2 guys said, they should really ship it with a stock attached to make it a "for sure" SBR, but I'm not the ATF.
It shouldn't make any difference about the stock adding to overall length, nor would it matter if one swaps a longer/shorter barrel onto a SBR'd AR. Both are considered "temporary" changes. As a dealer, I'd copy the exact lengths on the F3 over to the F4.
If the customer goes with different length parts and wants to notify ATF, they can note the change(s) in a letter after the transfer to them is complete. They should eventually receive a response that the change(s) were documented in the registry. It's not a requirement.
I wouldn't lose any sleep about the lack of a stock being an absolute "no-go," but I would mention it to Aero, unfavorably. Years ago, manufacturers sold SBR's as basically as complete lowers (only) with stocks, and when ATF decided this was technically not meeting the SBR definition (no barrel), the practice ceased, but nothing horrible happened. They'll probably correct Aero if/when they notice.