I understand this room even less than I've thought.
For the keycard:
Click here to view the secret text
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1. Someone or something has to press the one-use plate before the room is cleared. The construct won't do it because of the oremites; I don't have a pushing weapon so it can't be a keg, so it must be Beethro himself, unarmed.
2. When entering at (0,10) unarmed, I can extract one keg. I think that's where I have to enter to get the keycard.
3. When Beethro is unarmed, the only thing that can drop trapdoors is the construct, one at a time.
4. So I thought I have to blow up the builders, but that requires two kegs and I can't get a second one out.
5. If I could get a second keg, I would have to push it on the disabled firetrap, and then step on the pressure plate (or have the awakened construct step on it)
6. But for that I have to spend too much time in the top half of the room with the snake released, and as a result it inevitably gets stuck.
7. But even before all that, I can't get the construct out, for the same reason.