NOT SURE WHAT HAPPENED HERE, BUT THIS KIND OF JUST MAGICALLY FIXED ITSELF.
I made some changes to the Try/Catch structure to try to handle this specific exception and all of a sudden it went away.
The TRY block executes without ever setting an exception now.
Now the crazy part - I removed my changes - put it back to how it was, and it's executing.
Nothing else changed server side or client side, so I'm really at a loss here.
My best guess is that the add-in was a bad compile for my environment, and re-compiling it on my machine is what made it work.