I managed to solve this problem, although it came mostly through trial and error and figuring out the "right" order in which to do things. For those who want to avoid the bugs I faced, here are the results of the different approaches to taking on the basement:
Scenario 1: I crawled my way through, explored every nook and cranny, grabbed every bit of treasure I could. I saved before the secret door with the gas trap, defeated the trap, and allowed the two female thieves to join my party. Fought the evil party and won. Walked over to the trapdoor. THEN got the tiara. Didn't use the Knock spell to open the locked door (it can be bashed or picked). Went back to the trapdoor, and got stuck.
Scenario 2: Same as above, but this time I simply refused to let the thieves join. They walked away, I walked away. I then defeated the evil party, used the knock scroll to open the door, got the tiara, and THEN walked to the trapdoor. Still got stuck.
Scenario 3: This time, I didn't disarm the gas trap. Allowed the thieves to steal the sword. Went back to the earlier part of the basement and found them, fought them, killed them, and kept the sword. Went back to kill the evil party, then got the tiara. Then went to the trapdoor. Still stuck.
Scenario 4: This time I went back to a much earlier save, from before I'd beaten the evil cleric in the orc caves. I went back and defeated him, then returned to town. In town, I sold off all of the gear I'd found (I hadn't done so in previous saves, I think). I then went to the mansion, went through the dinner where you get the quest, accepted the quest, headed into the basement. I then went DIRECTLY to the guardian room where the sword is, killed the guardian, took the sword, got the magic items from the secret room, and left. I went directly to the coal room, disarmed the gas trap, went through, killed the two thieves outright, proceeded to the evil party encounter, killed them, used a knock scroll on the door, got the tiara, and then got a message saying I had both items so now all I needed to do was leave. Went to the trapdoor and climbed out. Problem solved.
I have two theories as to what's going wrong here.
Option 1 is that there's something screwy with the interaction between the Inv. system and the items you pick up. If you have the wrong stuff, or picked things up out of order, or maybe have older "Inv." items that you haven't sold yet, the checks for the two items get screwed up and you end up getting the message that you don't have them both. It's worth noting that the sword goes in as an "item" like any other (it's just a two-handed sword +2), but the tiara is an "inv." item. In looking at my "inv." in one of the bad saves, I noticed that I had a blank line between two items, which was still treated as something that could be selected. I'm wondering if maybe the problem doesn't come from that, and that the blank is supposed to be the sword or something. I dunno.
Option 2 is that it's something to do with the two thieves you meet, and that when you meet them, for whatever reason, ANY action taken that isn't just "Kill 'em on the spot" triggers some kind of event where the sword disappears. When you recapture the sword, I THINK it's classed as a different quest (the one with the tiara is, I think Quest 18, but reclaiming the sword is, I think, Quest 16). So, again, the only "right" way to proceed through this is to kill the two thieves on sight. No discussion, no "let them go," no "join the party," just "You see two thieves, they demand to know what you are doing, and you ...attack them at once!" or whatever the dialogue is.
I will say that the rest of the module, thus far, is pretty fun, although I'm wondering if there is some way to stealthily infiltrate the keep that I managed to screw up, or if you're supposed to always fight waves at the gate. Fun battles either way.