Let $A_i, i=1, \ldots, N$ be real (or complex) matrices of the same dimension. Let $r_i, i=1, \ldots, N$ be independent Rademacher random variables. The following inequality gives a bound on the expectation of the p-th Schatten norm: $$ E\left\|\sum_{i=1}^N r_i A_i\right\|_{S_{p}}^{p}\leq C\sqrt p \max\left\{\left\|\left(\sum_{i=1}^NA_iA_i^*\right)^{1/2}\right\|_{S_{p}}^{p}, \, \left\|\left(\sum_{i=1}^NA_i^*A_i\right)^{1/2}\right\|_{S_{p}}^{p}\right\}. $$ Question: whether some similar bounds known for p-th moment of the fixed q-th Schatten norm, i.e. something like $$ E\left\|\sum_{i=1}^N r_i A_i\right\|_{S_{q}}^{p}\leq C\sqrt p \max\left\{\left\|\left(\sum_{i=1}^NA_iA_i^*\right)^{1/2}\right\|_{S_{q}}^{p}, \, \left\|\left(\sum_{i=1}^NA_i^*A_i\right)^{1/2}\right\|_{S_{q}}^{p}\right\}. $$

If you do not care about the precise constants, the answer is yes, and this follows directely from the Kahane inequality (which says that, in any normed space and $1 \leq p,q<\infty$, the $L_p$-norm of $\sum r_i x_i$ is dominated by its $L_q$-norm, up to constant depending on $p,q$ only). The conclusion is that for every $1 \leq p, q <\infty$, there is a constant $C_{p,q}$ such that $$ \left(E \|\sum_{i=1}^N r_i A_i\|_{S_{q}}^{p}\right)^{\frac 1 p}\leq C_{p,q} \max\left\{\left\|\left(\sum_{i=1}^NA_iA_i^*\right)^{1/2}\right\|_{S_{q}}, \, \left\|\left(\sum_{i=1}^NA_i^*A_i\right)^{1/2}\right\|_{S_{q}}\right\}. $$

Observe that it is necessary to take the $1/p$-th power if you want to have $C_{p,p} \leq C \sqrt{p}$.

If you care about the constants, what you ask for is way too strong. It is clearly necessary to have $C_{p,q} \to \infty$ as $q \to \infty$ (as otherwise by considering diagonal matrices it would give Khintchine inequalities in $\ell_\infty$).