Unfortunately, though this works great for $k=1$, it can only be used for particular values of $\Omega$ when $k>1$. E.g., when $Y,Z$ have orthogonal columns, this can only be used for constant matrices of the form $\Omega = \alpha 1_k 1_k ^T$...

@fedja For $k=1$ there is a heuristic solution that works very well when $Y$ and $Z$ are drawn from Gaussian distributions: Order $Y$ and $Z$ on a linear combination of their values and independent Gaussian noise. It's relatively straightforward to figure out what coefficients for the linear combination will yield the desired value of $\Omega$. This is admittedly a far less general problem, but I imagine similar solutions are available for other forms of $Y,Z$.

Thanks, I figured this was the case :). However, your counterexample involves sums of singular non-symmeteric matrices, rather than SPD matrices, unless I'm missing something? Also, I wasn't referring to the Knutson and Tao solution, but rather the other comment in the linked answer: "Depending on what you want, there should be simpler results giving estimates on the eigenvalues of the sum. A book like Bhatia's Matrix Analysis might have some helpful material." So I was hoping their might be some nice trick :(