No, the B body rear will fit. You have to relocate the spring perches, but it'll fit. Of course you won't have the tire room you could, but I think that can be overcome with wheel selection. Plenty of people have used B body rears.
he is not going to get a 71-74 B body rear to fit under an early A body car, they are the widest 8 3/4” you can find. Well, except for a C body unit, which is very close in width to the one he has.
