The following is a development along Hari's answer.
As mentionned by Hari, we may assume that $B$ is an irreducible summand of size $m$. It inherits $q$-normality. Let $L$ be the sub-algebra of $M_m(\mathbb C)$ spanned by $B$ and $B^*$.
The subspace $\ker B\cap\ker B^* $ is invariant under $B$ and $B^* $. From irreducibility, we must have $\ker B\cap\ker B^* =\{0\}$. Set $H:=BB^* + B^* B$, which is Hermitian positive definite. From Cayley-Hamilton, $I_m$ is a polynomial in $H$, thus belongs to $L$. Therefore $L$ is unit algebra.
We prove now that $L$ is a simple algebra: let $J\ne(0)$ be a two-sided ideal in $L$. Choose $M\ne0$ in $J$. Then every $M^k$ with $\ge1$ belongs to $J$. By Cayley-Hamilton, we deduce $Tr(M)I_m\in J$. If $J$ is proper, this implies $Tr(M)=0$. Now, $M^* M\in J$ too, because $L$ is invariant under $C\mapsto C^*$ and $J$ is an ideal. Thus $Tr(M^*M)=0$, which is absurd. Therefore $J=L$ and $L$ is simple.
By Wedderburn's theorem, $L$ is isomorphic to $M_r(K)$ for some $r\ge1$, where $K$ is a division ring with $\mathbb C$ in its center. Because $L$ is finitely generated, $K$ must be of finite dimension over $\mathbb C$. From Frobenius' theorem, $K=\mathbb C$. Finally, $L$ is isomorphic to $M_r(\mathbb C)$.
At last, the morphism $M\mapsto M$ from $L$ into $M_m(\mathbb C)$ is non-zero. Because both $L$ and $M_m(\mathbb C)$ are simple, they must be isomorphic. In particular, they have the same dimension, therefore $L=M_m(\mathbb C)$.
By assumption, $\mathcal S_{2q}$ vanishes identically over $L$, thus over $M_m(\mathbb C)$. This implies easily $m\le q$.