Reporting back: easiest, most fun repair I've ever done.
I first wrote some notes about which color wires went to which terminals. Then I heated up the solder joints and the wires just popped off. Apparently they don't put the wires through the holes before soldering. Not what I'd call "best practice" but in this case it made my life a little easier and left me with pre-tinned leads.
Next I removed the nut from the new switch and put all the wires through it. Then I realized that the nut obviously would not fit through the hole in the hilt. I had the blade holder off anyway (had to resolder the resistor onto the LED module) so I just dropped the nut down her throat. I reached into the switch hole with an alligator clamp and grabbed the nut. From outside, I threaded a bit of wire through the nut so I wouldn't lose it.
Then I threaded all four wires through the nut and out the hole. I grabbed the four wires with the gator, slipped the wire handle off the nut and let it fall back into the hilt. There was not enough bare wire to make a proper joint, so I stripped another 1/8 or so off each one. Then I cut a bit (too much, you'll see why) of heatshrink for each wire and put the wires throug.
Finally, I poked each wire through its respective terminal and was able to wrap each one all the way around (except the red one, which had enough bare that I didn't bother to strip it). Having only two hands, soldering isn't something I'm physically capable of but I made it work by pre-tinning the iron itself. Since the wires were also tinned, it was a breeze.
Once the joints were all soldered, it was time to shrink the wrap. Retrieving the heatshrink from further up the wire was challenging and I would not have been able to do it without the extraordinary tweezers I recently purchased from Sparkfun.com. I don't know if this was a good idea but I used the soldering iron to heat the heatshrink.
Now time to thread the nut. Hmm. How am I going to get my fingers or any kind of tool down the throat of this thing to twist that nut? Good old (and I do mean OLD) needlenose. I was lucky in that when I pushed the switch into the hole, the nut lined right up with the threads. So I was able to gently push it along with the end of my needlenose.
Oh right. Too much heatshrink. It turns out I had used enough heatshrink on two of the wires that I couldn't push the switch all the way into the hole. So I did the dumbest thing I possibly could. I sliced it. I didn't cut around the wrap, slicing into two sections that could bend like an elbow. No. I cut from the end, up the sleeve. It worked; I got the switch in but now I'm worried the wrap will gradually pull apart until that cut goes all the way along.
And that was that. Like I said: easio, cheapest repair I've ever done. I didn't wind up putting any connectors in. Given the amount of effort this took, the connectoes would have been more work than they're worth. I am going to do my Warrior Acolyte with them however, since it's going to be upgraded in stages and I'd rather not have to unsolder and resolder my own work.
For my first time wielding a soldering iron, I think I did damn well. Time will tell.