You have simply reproduced the classical analysis of the MMX, which
I have also given in my article "There is no length contraction".

You should take into account Einstein's postulate, according to which
the speed of light is the same in all frames.
Then, automatically, you get x' = ct', which is the length of the
"longitudinal" arm when the apparatus is moving in the "ether".
But, also, according to special relativity, y' = ct', which is the
corresponding length of the "lateral" arm.
The two lengths being equal, no fringe displacement can be observed.