No. You can get a higher $I(U;V)$ using asymmetric channels. Below I construct a counterexample, but first a restatement.

To summarize your question, there is an input $U$ distorted by three independent binary hops, each described by $2\times 2$ stochastic matrices $C_L,\ C,\ C_R.$ Labeling all your RV's, $$U\overset{C_L}{\to}X\overset{C}{\to}Y\overset{C_R}{\to}V.$$ 
We are interested in maximizing $I(U;V)$ 
 subject to constraints: 

- $C$ is determined by nature.
- $C_L$ is such that $I(U;X) \leq r_L$, 
- $C_R$ is such that $I(Y;V) \leq r_R$,
- $U$'s distribution is such that the left channel's output (i.e. $X$) is $B(1/2)$.

------

You claim the $C_L,C_R$ which produce the maximum are binary symmetric. 

* If $C_L$ is a BSC with $B(1/2)$ output then its input must also be $B(1/2).$ For a given rate $r_L \leq 1$, then there is at most one 'positive' (i.e. can't be improved by relabeling the outputs) BSC $C_L$ whose output is $B(1/2).$  
* You have assumed $C$ is a BSC, so with a symmetric input its output is also symmetric. 
* For a rate $r_R\leq 1$ there is only one positive choice for $C_R.$

So to say they are binary symmetric is to determine all of $U,C_L$ and $C_R$.

Now take $C$ a perfect channel, 
$C= \left[\begin{smallmatrix}
    1 & 0 \\
    0 & 1
    \end{smallmatrix}\right]$ and $r_L=r_R=0.4.$ The associated positive BSC for this rate has crossover probability $\approx 0.15,$ and the end-to-end mutual information can be computed: 
$$I(U_{BSC}, V_{BSC})< 0.161$$

However, trying randomly[1] I have found $U^\ast, C_L^\ast, C_R^\ast$ that satisfy all the mutual information constraints, but have greater $U$-to-$V$ mutual information. They happen to be close to a Z-channel:

\begin{equation}
   I(U^\ast; V^\ast) > 0.17,
\end{equation}
\begin{equation}
   C_L^\ast \approx \left[\begin{smallmatrix}0.713 & 0.287 \\ 0.0055 & 0.9945 
    \end{smallmatrix}\right], \qquad
    C_R^\ast \approx \left[\begin{smallmatrix} 0.7099 & 0.2901 \\ 0.0485 & 0.9515 \end{smallmatrix}\right], \qquad
   U^\ast \approx \left[\begin{smallmatrix}0.6989 & 0.3011 \end{smallmatrix}\right].
\end{equation}

----

A gentle introduction for a good, visualisable framework for studying binary symmetric channels is given in a short paper, *Algebraic Information Theory for Binary Channels* by Martin, Moskowitz and Allwein. Under this framework your maximization can be restated as a convex optimization problem for which I see no easy special cases. There might be some magical connection to KL divergence which I am not seeing.

An easier-to-investigate (and arguably more interesting)  problem is one identical to yours but omitting the fourth constraint, that $X\sim B(1/2)$. 

-----

[1]: Below is a crude counterexample finder. 

    % Helper functions
        % Binary entropy
    fn_h = @(p) -p.*log2(p) - (1-p).*log2(1-p); 
        % MI across mtx_bc when v_distn is input
    fn_I = @(mtx_bc,v_distn) fn_h(v_distn(1)) + fn_h(v_distn*mtx_bc(:,1)) ...
        - sum(sum(-log2(diag(v_distn)*mtx_bc).*(diag(v_distn)*mtx_bc)));
        % Channel matrix when P(out=0|in=0)=pa, P(out=0|in=1)=pb
    fn_mtxBC = @(pa,pb) (pa>=pb)*[pa, 1-pa; pb, 1-pb]; fn_p = @(mtx) mtx(:,1);
    
    % Set params
    d_r_L = 0.4; 
    d_r_R = 0.4;
    d_xp = 0.146102; % solution to 1-H(p) = 0.4
    mtxBSC = fn_mtxBC(d_xp, 1-d_xp);
    
    % Search 
   	while true
		mtxL = mtxBC(rand, rand);
		mtxR = mtxBC(rand, rand);
		v_d = (mtxL'\[0.5, 0.5]')';
		if (abs(sum(v_d)-1) > 0.001 || ...
			min(v_d) < 0)
			continue
		end
		fprintf('+');
		if(fn_I(mtxL, v_d)      > 0.4 || ...
		   fn_I(mtxR, v_d*mtxL) > 0.4)
			continue;
		end
		if fn_I(mtxL*mtxR, v_d) > fn_I(mtxBSC*mtxBSC, v_d)
			break;
		end
	end