The first problem you pose never has a solution. The matrices of the form $\mu^{\otimes 2}$ for nonzero $\mu$ are the extreme rays of the positive semidefinite cone. That is to say, your condition on the second moments implies that $\mu^{\otimes 2}$ is a scalar multiple of $\mu_i^{\otimes 2}$ for some $i$. The normalization then gives $\mu = \mu_i$.

It is a standard result that the matrices of the form $\mu^{\otimes 2}$ for nonzero $\mu$ are the extreme rays of the positive semidefinite cone. That is to say, your condition on the second moments implies that $\mu^{\otimes 2}$ is a scalar multiple of $\mu_i^{\otimes 2}$ for some $i$. The normalization then gives $\mu = \mu_i$. So the there never exists a solution to the main question.