They do make thicker C clips for GMs though.
Thinking on it, I'd wouldn't use peel-able shim stock behind the C-clips as they'd surely give way and start bouncing around your diff case.
You could have someone push in on the axle while you measure the gap behind the c-clip left and right.
Using hardened stock or hardened steel washers have them drilled and milled the correct thickness.
https://www.mcmaster.com/hardened-washers