By "unsolvable" I mean that you're screwed over in final outcomes, not that TDT fails to have an output.

Oh ok. So it's unsolvable in the same sense that "Choose red or green. Then I'll shoot you." is unsolvable. Sometimes choice really is futile. :) [EDIT: Oops, I probably misunderstood what you're referring to by "screwed over".]

The interesting part of the problem is that, whatever you decide, you deduce facts about the background such that you know that what you are doing is the wrong thing.

Yes, assuming that you're the sort of algorithm that can (without inconsistency) know its own choice here before the choice is executed.

If you're the sort of algorithm that may revise its intended action in response to the updated deduction, and if you have enough time left to perform the updated deduction, then the (previously) intended action may not be reliable evidence of what you will actually do, so it fails to provide sound reason for the update in the first place.