This is a critical part. The rest we can hash out.
It seems that if neither party can know that they are moving, then you have symmetry between their perceptions. There can be no preference involved. All we have (now) is when he pressed the button the train was 6 μls away from the first clock.
If point A sees point B to be 6 μls away, how can point B see point A to be a different distance away? The speed of travel is the same for both of them. One can't have preference such as to be shorter or longer regardless of any clocks. And any calculation that would apply to one frame would equally apply to the other's.
It could have equally been stated that the first stop clock was 6 μls away from Einstein when the button was pressed.
That one number is the only thing actually affixed in the scenario. Everything else is a frame dependent calculation. You have to have something with which to start that is common for both frames. It makes more sense to me to make that first clock x = 0 = x'.