After a few tries at the EHUP to no avail, I decided that the pont of the puzzle was that it was impossible, and you just need to let some things go. Feeling satisfied, I went downstairs with the doctor and completed the story. Only after I looked at the badge list did I realize I'm just bad at puzzle games.
In retrospect, the survivors should've spent more time trying to escape their apocalyptic wasteland and less venturing forth to put down crosses and clear the wreckage of their fallen comrades.