One further idea:
On page one you posted a video showing an add-on to the head of the key. Is it possible the round two pronged thingy slides over the head of the make-up key? Note that the prongs are also the same size as the keyway, such that after the key is finished with all bits added, these two prongs would engage the top of the keyway once they slide over at right angles.
I am now certain that the two pronged key with the large round head does fit over the top of the make up, square headed key. Imagine it sliding over at a 90 degree angle. The inside of the prongs are squared off to fit the square groove, and the bottom of the prongs fit the keyway in line with the bitting that has been added.