So I experimented once again trying to get these stupid little wires soldered. I got my iron down to 400F but it was barely enough to melt the lead solder. I had to go up to 500F to try the lead-free solder. The lead free solder was flux cored and that seemed to tin the wire much better, but the wire still burned back some. I switched to lead solder and I was able to get the wire to bond with the strand, finally!
Unfortunately during the process I dropped hot rosin on the diaphragm and it melted a hole through it.
It was a good experiment though. I have one more capsule with no continuity that I can try this on.
Sad news for those who are following this thread. The last dead capsule was truly dead. I found that the wire from the capsule to the post was broken, something I think I might have done when getting the humbucking coil basket off the diaphragm, and I spent a good part of an hour gently cleaning the glue and such off the post and wire. I was able to tin the wire and get it soldered to the post..
Only to find that it still didn't have continuity.
After messing with it and double checking both sides, I decided to take the diaphragm off the base only to find that the wire was broken at the coil itself as well. There's no fixing that.
I *might* try taking the coil off the melted diaphragm and swapping it to this diaphragm at some point although I currently don't know how to do it. I guess I have nothing to lose trying.
So I tried something crazy. I swapped the coils from one diaphragm to the other. I was able to easily remove the coils and unwind a half turn of both leads to get enough wire to solder. I took the good diaphragm and was able to tack the other coil to it, but I had a really hard time getting it lined up on the crease where the other coil was removed. I think part of the problem was the remaining glue and part of the problem was that the coil distorted slightly when being removed. Even working as gently as possible I could only barely get it lined up. I was also not able to get it perfectly aligned in the magnetic gap but I can't seem to notice any rubbing. The gap is fairly wide, maybe about 2x the thickness of the coil, which helped.
Well, the swapped coil capsule works. It doesn't sound a good as the others but it works at least. It's surprisingly close though. I might try reseating the diaphragm at some point but right now it's not a priority.