It may be your best friend, but the companion cube is still a cube - and that means we can use it to break the puzzle.

When you reach the incinerator room, drag the cube against the wall so that it's slightly tilted, then position it against the door as seen below. Save the game, then stand on the button for a second to open the door - when you get off the button, the cube will get stuck between the doors.

Stand on the button again, and if you're lucky, the cube land right behind the door and trigger the unique dialogue. If it gets stuck near the standing button, however, nothing will happen and you'll have to load your save.

Nope! You'll get the usual softlock dialogue and the door will be permanently forced open, but you still need to euthanise your companion cube to proceed. :(