Fock operator invariance under unitary transformation
The exchange operator is not a operator by itself, it is only defined with the orbital it is working on: 
$%
\newcommand{\ll}{\left\langle}\newcommand{\rr}{\right\rangle}
\newcommand{\lb}{\left|}\newcommand{\rb}{\right|}
\newcommand{\op}[1]{\mathbf{#1}}$
$$\begin{align}
  &amp;&amp; \op{F}_i &amp;= \op{H}^\mathrm{c} + \sum_j (\op{J}_j - \op{K}_j),\\
\text{with}&amp;&amp; 
     \op{J}_j\lb \phi_i\rr &amp;=
       \ll \phi_j(\op{x}_1) \rb r_{12}^{-1} 
       \lb \phi_j(\op{x}_1) \rr \lb \phi_i(\op{x}_2) \rr,\\
\text{and}&amp;&amp;
     \op{K}_j\lb \phi_i\rr &amp;=
       \ll \phi_j(\op{x}_1) \rb r_{12}^{-1} 
       \lb \phi_i(\op{x}_1) \rr \lb \phi_j(\op{x}_2) \rr.
\end{align}$$

While in the Coulomb operator case the $\lb \phi_i\rr$ doesn't do anything, so you can take it through the proof as a constant:*
\begin{align}
  \sum_j\op{J}_j\lb \phi_i\rr 
  &amp;= \sum_j 
     \ll \phi_j(\op{x}_1) \rb r_{12}^{-1} 
     \lb \phi_j(\op{x}_1) \rr \lb \phi_i(\op{x}_2) \rr\\
  &amp;= \sum_k \sum_l \sum_j U_{kj}^*U_{lj}
     \ll \phi_k(\op{x}_1) \rb r_{12}^{-1} 
     \lb \phi_l(\op{x}_1) \rr \lb \phi_i(\op{x}_2) \rr\\
  &amp;= \sum_k \sum_l \delta_{kl}
     \ll \phi_k(\op{x}_1) \rb r_{12}^{-1} 
     \lb \phi_l(\op{x}_1) \rr \lb \phi_i(\op{x}_2) \rr\\
  &amp;= \sum_k 
     \ll \phi_k(\op{x}_1) \rb r_{12}^{-1} 
     \lb \phi_k(\op{x}_1) \rr \lb \phi_i(\op{x}_2) \rr\\
  &amp;= \sum_k \op{J}_k \lb \phi_i\rr
\end{align}

That is not the case in the Exchange operator, as it switches the orbitals:
\begin{align}
  \sum_j\op{K}_j\lb \phi_i\rr 
  &amp;= \sum_j 
     \ll \phi_j(\op{x}_1) \rb r_{12}^{-1} 
     \lb \phi_i(\op{x}_1) \rr \lb \phi_j(\op{x}_2) \rr\\
  &amp;= \sum_k \sum_l \sum_j U_{kj}^*U_{lj}
     \ll \phi_k(\op{x}_1) \rb r_{12}^{-1} 
     \lb \phi_i(\op{x}_1) \rr \lb \phi_l(\op{x}_2) \rr\\
  &amp;= \sum_k \sum_l \delta_{kl}
     \ll \phi_k(\op{x}_1) \rb r_{12}^{-1} 
     \lb \phi_i(\op{x}_1) \rr \lb \phi_l(\op{x}_2) \rr\\
  &amp;= \sum_k 
     \ll \phi_k(\op{x}_1) \rb r_{12}^{-1} 
     \lb \phi_i(\op{x}_1) \rr \lb \phi_k(\op{x}_2) \rr\\
  &amp;= \sum_k \op{K}_k \lb \phi_i\rr
\end{align}



* Let 
$$
\mathbb{U}=\left(\begin{matrix}
  U_{11} &amp; U_{12} &amp; \dots \\
  U_{21} &amp; U_{22} &amp; \dots \\
  \vdots &amp; \vdots &amp; \ddots\\
  \end{matrix}\right);
\mathbb{U}^\dagger\mathbb{U}=\mathbb{E}
\implies U_{ij}^* U_{kl}= \delta_{ij} = \begin{cases}
   1; &amp; i = j = k = l \\ 0; &amp; i \neq j \dots\\ \end{cases}
.$$