ALL HAIL...............................um, something?
Although, really, I'm not entirely sure what it was that did it.
In case anybody else has the same problem, here's what I did:
1. I reloaded from a game before I entered the Factory Floor module for the first time. (I get paranoid about losing stuff or whatever, and tend to save every like three minutes. In different slots. And it's definitely worth it.)
2. Then, I walked straight to HK-47, not talking to anybody else in the module, and received his task.
3. I had already figured out which of the droids were spies, so I took each one out (the Factory Floor one, then the Reactor module one, then the Landing module one), still without talking to anybody.
4. I went straight back to HK-47, not talking to anyone.
And, somehow, it worked.
Maybe all I needed to do was start from before entering the module for the first time, but I thought it would be annoying if that turned out to not be it and have to try other things too. So I just restarted from before the module, then didn't do anything else I thought might even remotely conceivably mess it up.
Oh, and HK-42: I've also noticed that the sounds in battle aren't always quite right. People or creatures groaning or yelling or droids squealing when they get hit or die don't always seem to match. But I don't find it that big a deal. ^_^