General question first: upper/lower bound a sum of Kronecker products by its components. More specifically, how is $$ \Vert\sum_{\alpha}S_{\alpha}\otimes B_{\alpha}\Vert$$ bounded by the operator norms of $d_{S}\times d_{S}$ dimensional matrices $\{ S_{\alpha}\}$ and $d_{B}\times d_{B}$ dimensional matrices $\{ B_{\alpha}\}$? The operator norm $\Vert\cdot\Vert$ denotes the (usual) max abs singular value norm. The symbol $\otimes$ denotes the Kronecker or tensor product of matrices. Do we have a Cauchy-like inequality?

More specific question:

We have a "basis" set $\{ S_{\alpha} \}$
of $d_{S}^{2}-1$ *Hermitian and traceless* matrices: they
satisfy $$\text{Tr}(S_{\alpha}S_{\beta}^\dagger)=d_{S}\delta_{\alpha\beta}$$
where $\delta_{\alpha\beta}$ is the Kronecker symbol and
are all normalized with respect the operator norm: $\Vert S_{\alpha}\Vert=1$.
In short, they would be an orthonormal basis for the Hilbert-Schmidt
inner product if the identity matrix was included and they were normalized properly. For $d_s=2$ the $\{S_\alpha\}$ can be taken to be the Pauli matrices.
The matrices $B_{\alpha}$ are also Hermitian and traceless but otherwise
arbitrary. I would like to bound $\Vert\sum_{\alpha}S_{\alpha}\otimes B_{\alpha}\Vert$
from up and down by a factor of $\max_{\alpha}\Vert B_{\alpha}\Vert$.
The best I have done so far is rather miserable:$$
\frac{1}{d_S}\max_{\alpha}{\left\Vert B_{\alpha}\right\Vert }\leq\Vert\sum_{\alpha}S_{\alpha}\otimes B_{\alpha}\Vert\leq(d_S^{2}-1)\max_{\alpha}{\left\Vert B_{\alpha}\right\Vert }$$

The closest reference that I have found on this material is the work by Chansangiam et al [ScienceAsia 35, 106 (2009)] which I am still going through. This question is related to a question in physics for bounding fidelity of evolutions in open quantum systems. In short I want to bound the maximum energy scale in a combined system with that of its components.