In fact, it’s even better than that — looking at your matrix, it’s actually a linear eigenproblem in -k'^2 (assuming all of your \eta_\ell factors are constants): if you just move the k'^2 terms to the other side, the equation you wrote takes the form Bc = -k'^2 c, so you can just use sqrt.(.- eigvals(B)) to find the roots k'.
If your matrix is 7 \times 7 as shown in your post, you should be able to solve 1000 of these eigenproblems in a fraction of a second.