  We study the eigenvalue distribution and resolvent of a Kronecker-product
random matrix model $A \otimes I_{n \times n}+I_{n \times n} \otimes B+\Theta
\otimes \Xi \in \mathbb{C}^{n^2 \times n^2}$, where $A,B$ are independent
Wigner matrices and $\Theta,\Xi$ are deterministic and diagonal. For fixed
spectral arguments, we establish a quantitative approximation for the Stieltjes
transform by that of an approximating free operator, and a diagonal
deterministic equivalent approximation for the resolvent. We further obtain
sharp estimates in operator norm for the $n \times n$ resolvent blocks, and
show that off-diagonal resolvent entries fall on two differing scales of
$n^{-1/2}$ and $n^{-1}$ depending on their locations in the Kronecker
structure.
  Our study is motivated by consideration of a matrix-valued least-squares
optimization problem $\min_{X \in \mathbb{R}^{n \times n}}
\frac{1}{2}\|XA+BX\|_F^2+\frac{1}{2}\sum_{ij} \xi_i\theta_j x_{ij}^2$ subject
to a linear constraint. For random instances of this problem defined by Wigner
inputs $A,B$, our analyses imply an asymptotic characterization of the
minimizer $X$ and its associated minimum objective value as $n \to \infty$.
