So after taking a moment to think (what a concept), I realized the piece of double-sided tape that saved my bacon was perfect for what I needed.
I placed a spacer on the tape, cut an outline around the spacer, then pressed the second spacer of the stack onto it. Kinda looked like an oreo.
I did this for all 4 stacks. Then put another piece on the bottom of each stack.
I punched holes through the stack with a Philips screwdriver, used the screws as placement guides, and pressed the stacks onto the monitor over the VESA holes.
Worked perfectly.
Notice that the lower part of the arm is disconnected, and only the upper is attached to the monitor. I didn't know it would do that, but in all the wrangling it just fell off. Duh. Makes it a lot easier to install on the monitor...