I've never had success soldering wires before. I read up on it and decided a torch would work best for small automotive bullet connectors. This Bernzomatic unit works very well. It was worth every penny I don't have.

The climate control illumination wiring is interesting; on my car the power from the lighting potentiometer exits at spot 15 on the connector through a grey wire with a violet stripe and makes its way down to the emergency flasher switch. I guess it's supposed to come on with the lights, but the bulb inside must be burnt out. Interestingly, the shift plate illumination is full power and doesn't dim with the potentiometer. Then a grey wire with a blue stripe branches off of the lighting wire for the emergency flasher switch and lights the four climate control bulbs.
On a car with manual climate control, the wire exiting the instrument panel is grey with a green and a violet stripe and it goes straight to the climate control lighting (which also has four bulbs plus one inside each dial switch). Then a grey wire with a violet stripe goes from it and lights the emergency flasher switch.
Removing the automatic climate control illumination wiring from my car and then adding the wiring from the manual climate control would have been simple were it not for a few inches of the wire where it connects to the emergency flasher switch being cut off, and even if it was intact, feeding it with the connector attached would have probably been impossible due to how tight the wires are squeezed into the sheathing.
I cut the grey wire with violet stripe on my car up by where it plugs into the instrument panel. I decided that I would remove the wire which had been cut on the manual harness, and solder the wire from my car in its place, since it was the same color and already inside the sheathing going to the emergency flasher switch. I heated up the bullet connector with the torch so the solder would melt and then pulled out the wire I didn't need, and then inserted the wire coming from the emergency flasher switch.
I drilled a hole in a piece of wood and that did a good job of holding the connector still while I heated it up and swapped the wires. The end result was the harness ending up exactly as it should have been, and with the added benefit of not having to thread the wire through the sheathing on the way to the emergency flasher switch.

My first attempt using the torch was successful, and it looks just as good as it did before. A bad situation has been corrected! I am very happy that I didn't take the easy way out and use a butt connector.

The plastic covering installed over the connectors.

The feed wire going from the instrument panel to the climate control illumination wiring replaced the wire that I cut and soldered between the illumination wiring and the emergency flasher switch. It is wired correctly and now there is a grey wire with a blue stripe left over branching out of the emergency flasher switch. I was going to remove it but then had an idea--I can run it up to the instrument panel and use it to light a temperature display which I can install from a W124.

As of right now, the climate control wiring is functional. The blower motor works on all speeds, the dial and faceplate illumination work, and the compressor gets power when the air conditioning dial is turned on. I still have some parts of the wire harness to remove, but electrically the car is about ready to have the air conditioning charged with R-12. I still have to install the hoses, though.