Non-stupid solution added by changing a single block. I hope.
Which block?
The block to the right of the yellow button in the bottom left is now red, where it was gray.
Anyways, solutions!
KK: 1 left, onto yellow.
Al: 2 up, 1 left, onto green.
KK: 1 down, 2 left, 1 down, onto green.
Al: 1 left, 1 down, 1 left, onto yellow.
KK: 2 left, onto yellow.
Al: 1 left, 2 up
KK: 1 up, onto green
Al: 2 left, 3 down, onto blue
KK: 1 down, 2 left, 1 up, onto green
Al: 3 up, 3 right, onto yellow
KK: 1 down, 2 left, 1 up, onto yellow
Al: 1 left, 2 down, 1 right, onto yellow
KK: 1 down, 2 right, 1 up, onto green
Al: 1 right, 1 up, 2 right, onto red
KK: 1 up, 1 right, 1 up, onto green
Al: 1 left, 1 down, 1 left, onto yellow
KK: 1 up, 2 left, onto yellow
Al: 1 left, 2 up, 1 right, onto yellow
KK: 2 right, 1 down, onto green
Al: 3 left, 2 down, embrace KK, 1 down, onto blue
KK: 2 up, 2 left, 1 down, onto yellow
Al: 2 up, 2 left, 1 up, 1 left
KK: 1 up, onto red
Al: Push ball down 2, push ball left 1 and out the door.
Al: Standing on the threshold of the door, recapture KK in the Illborn Exchange
If the whole recapturing KK from halfway to the next room thing doesn't work, I think this might still be impossible, if we want to keep all 3 things. We need to have red depressed in order to exit the door, and we only have three things capable of pressing red: Al, KK, and the ball. Pressing red from the tile next to the door is impossible. Therefore, without the introduction of a fourth button-pushing element, if we can't use the IE to get KK here, completing the puzzle with all three in tow is impossible