I am beginning to think that maybe you are looking at the wrong cross member.
There is a very small cross member at the very end of the C3 frame (J) that the gas tank straps lock onto and some of the bumper brackets connect to, but the cross member that is being worked on is the cross member farther forward (K and H) that supports the rear differential and spring.
I don't believe that either cross member could allow the bumpers to support the frame though. Maybe on other car designs with a full width rear bumper, but not on a car with 2 independent rear bumper like the Corvette has.
In my opinion, the rear bumpers on both the C2 and early C3 Corvettes were pretty much decorations, not real bumpers. Almost any small hit on the bumpers resulted in significant body damage where the bumpers bolted to the body.
That was also why I had practically Zero concerns when I eliminated the rear bumpers from my '68 when the rear was re-designed.