Okie dokie I was waiting for 7810 voltage regulators for ages, I'm still waiting on them but I am impatient so I ordered some LM2940T voltage regulators (datasheet). These are a 10V, 1A regulator and it's the same TO-220 package. It's a fancier, more expensive voltage regulator that is overkill in this application but the 7810s I ordered aren't going to be delivered until well into 2023.
Anyway I'll let the photos tell the story. If you have an early 240 cluster, a GT cluster, or any cluster that has this style of regulator and you would like it rebuilt using a modern semiconductor regulator, drop me a DM. I will do it for you for $50 including return postage.
Old internals (broken):
New regulator + thermal paste + wiring pigtails:
Bolted in place. I used blue loctite on the M3 screw. I drilled the hole in the housing.
All wired up
After taking the car for a drive, gauge is sitting exactly where I want it. I measured the regulated output at 10.6V.