If the gun has BATFE approval, he's good to go.
The original wording of his ad was what set my spider sense tingling. You have to admit he made it sound like he's selling the gun, plus everything you need to make a buzzgun out of it.
The thing everyone's been calling a "lower receiver" is a trigger pack, and is not the "receiver", as defined by the BATFE. I'd imagine, given the similarity of construction between the StG-44 and its lineal descendant, the G-3, that legal precedents established for the HK rifle would apply here. As long as adding the FA trigger group won't make it a buzzgun, there should be no legal problem, but you're out in a gray area with a lot of these parts kit guns that rely on BATF opinion letters.
"Consrtuctive possession" means what you say it does, and has been used by the BATFE to describe the situation where one, say, owns an AR-15 and all the M-16 fire control bits. Even if they are not in the gun, the BATFE considers you to be in "constructive possession" of a machine gun.