Simplification of $x = \left(A + \sum_i \alpha_i B_i\right)^{-1} A \ y$, $A \succ 0, A \in M_{n,n}(\mathbb{C})$, $\textrm{rank}(B_i) = 1$

Question

I have a matrix inversion problem on hand. I want to reduce the matrix inversion complexity, if at all feasible. Let me give a brief overview of the problem definition.

Problem definition: Say $A \succ 0, A \in M_{n,n}\left( \mathbb{C} \right)$, is Hermitian and full-rank (positive definite) matrix. $B_i \in M_{n,n}\left( \mathbb{C} \right)$ and rank-$1$, $\alpha_i \in \mathbb{C}$ some constant, $y \in M_{n,1}\left( \mathbb{C} \right)$ some column vector.

\begin{align} x = \left(A + \sum_{i=0}^m \alpha_i B_i\right)^{-1} A \ y \end{align}

Question: Can this matrix inversion be simplified such that big matrix inversion can be avoided (since $n \approx 8$k)? Thank you in advance.

I can't make any progress with matrix inversion lemmas, e.g., in matrix cookbook.