I was able to get that nut off and the box out without removing the heater control or drilling any holes.
I used this ratchet with a shorter socket and was able to get it loose. I did the rest by hand with a magnet handy and am confident I'll be able to get it back on.
https://www.amazon.com/dp/B00DSR50BQ..._v5RsDb53324CF
Tonight, I'll clean up the heater box and add new padding to both sides of the door/flap, as well as 3M caulk strips where the core will sit. The heater control is going to get lubed up VERY well, too and should be good for another 50 years.
Also, during reinstall: I think leaving the top of the heater box leaning toward the back of the car (and the cables unscrewed from their respective colored tabs) makes it easier to reapply the cables to their correct mounts. Unscrewing them from the colored tabs also releases tension, limiting the chances of actually breaking the tabs.
Stay tuned....