Why don't we transpose $\delta^{l+1}$ in back propagation?
You have some wrong dimension here. Rules  for Dim of weight $W^{[l]} = d^{l} * d^{l-1} $
$ W_0 = [ 4 * 2 ] $
$ W_2 = [ 2 * 4 ] $
As
$  dim (z^{[2]})  = [2 * 1] $
so is
$ \delta^{[2]}$
So
$ W_{2}^{T} \delta^{2} $ is $ [4 * 2 ] * [2 * 1] = [4 * 1 ]  dimension $