In deriving Special Relativity equation, from math point of view, why don’t we chose length perpendicular to the line of motion to be modified? In deriving Special Relativity equation, we modify the length parallel to motion in a way that outside observer would agree that light moving in perpendicular and in parallel with motion would meet in the same time and spot (the same event). But the same effect could be achieved if instead we make the length perpendicular to motion becoming longer and set the length parallel to motion fixed.
I’m referring to this derivation from Feyman Lectures on Physics’ The Special Theory of Relativity (https://www.feynmanlectures.caltech.edu/I_15.html):

The first equation in question is:
$t_1 + t_2 = \frac{2L/c} {1 - u^2 / c^2}$ (Eq. 1)
which is the time needed for light to go from $B$ to $E^\prime$ and coming back to $B^\prime$.
The second equation is:
$2 t_3 = \frac{2L/c} {\sqrt{1 - u^2 / c^2}}$ (Eq. 2)
which is the time needed for light to go from $B$ to $C^\prime$ and coming back to $B^\prime$.
In order to make it equal so the light would arrive at $B^\prime$ in the same time, we choose to shorten the length $L$ parallel to line of motion to $L \sqrt{1 - u^2/c^2}$, so Eq. 1 would become:
$t_1 + t_2 = \frac{2L/c} {\sqrt{1 - u^2 / c^2}}$ (Eq. 3)
where $t_1 + t_2$ would equal with $2 t_3$ from Eq. 2.
But the same effect could be achieved by lengthening line $L$ perpendicular with line of motion to $L / \sqrt{1 - u^2/c^2}$ so Eq. 2 would become:
$2 t_3 = \frac{2L/c} {1 - u^2 / c^2}$ (Eq. 4)
where again $t_3$ would equal with $t_1 + t_2$ from Eq. 1.
If we are doing that, what I know immediately from the equation is the time in the other observer would be slowed down by a factor of $\frac{1}{1 - u^2 / c^2}$ instead of $\frac{1}{\sqrt{1 - u^2 / c^2}}$. 
I know there would be paradoxes when it does happen, and it would not be invariant with Maxwell’s equation, but just from Special Relativity point of view, what is wrong with this? Why don't we choose this solution?
And please do not just vote down this question, but also give reason when it's considered stupid. I have searched and no answer to find.
 A: 
just from Special Relativity point of view, what is wrong with this? Why don't we choose this solution?

This transform is called the Voigt transform. It may have actually been developed prior to the development of the Lorentz transform. While the Voigt transform does respect the second postulate, the invariance of c, it does not respect the first, the principle of relativity. 
Specifically, the principle of relativity requires that $\Lambda^{-1}(v)=\Lambda(-v)$ where $\Lambda$ is the transformation matrix between inertial frames. But with the Voigt transform the inverse of the Voigt transform is not even a Voigt transform. This means that two frames would be distinguishable from one another since one would transform differently than the other. 
A: As you have pointed out, your new transformation culminates in paradoxes like this SE question to which you also gave an answer. If it were not paradoxical, it could be a challenging point of view.
I have published an article regarding this matter in a dissenting journal, though a small part of the article is devoted to this matter. (To download the article, please click on "Complete Issue (.pdf)" for "Summer 2018. Volume 29. Special Issue 3" in this link, and see p. 45. Sec. 4.1.)   
