Consider the sum
$$J=  \sum_{m_3=-j_3}^{j_3} \left|C^{j_3,m_3}_{j_1, m_1; j_2, (m_3 - m_1)} \right|^2$$
with $j_1,j_2,j_3\in\mathbb{N}$ and $m_1\in\mathbb{Z}$. For an nonvanishing sum we also need $|j_1-j_2|\leq j_3\leq j_1+j_2$ and $-j_1\leq m_1\leq j_1$. With these restrictions the sum is independent of $m_1$, I may set $m_1=0$. The sum also increases mononically with increasing $j_3$, reaching its maximal value $J_{\rm max}$ for $j_3=j_1+j_2$, hence
$$J_{\rm max}=\sum_{m_3=-j_2}^{j_2} \left|C^{j_1+j_2,m_3}_{j_1, 0; j_2, m_3} \right|^2$$
$$\qquad=\sum_{m_3=-j_2}^{j_2}\frac{4^{j_1} (2 j_2)! (2 j_1+2 j_2+1) \Gamma \left(j_1+\frac{1}{2}\right) (j_1+j_2-m_3)! (j_1+j_2+m_3)!}{\sqrt{\pi }(1+2j_2) j_1! (2 j_1+2j_2+1)! (j_2-m_3)! (j_2+m_3)!}.$$
This sum may be evaluated in closed form, in terms of a hypergeometric function,
$$J_{\rm max}=\frac{2\Gamma \left(j_1+\frac{1}{2}\right) \Gamma \left(j_2+\frac{1}{2}\right) (j_1+j_2+1)! \bigl[ \, _3F_2(1,-j_2,j_1+j_2+1;-j_1-j_2,j_2+1;1)-1/2\bigr]}{\sqrt{\pi }(1+2j_2) (j_1+1)! (j_2+1)! \Gamma \left(j_1+j_2+\frac{1}{2}\right)}.$$