Find a maximum matching that saturates a given set of vertices
After some research, I found out that my question is a special case of the problem of priority matching. In this case there are two priority classes, $X_0$ and $X_1 := V\setminus X_0$. The goal is to find a matching that maximizes the number of saturated vertices in $X_0$, and subject to that, the number of saturated vertices in $X_1$. There are efficient algorithms that solve this problem for any number of priority classes. It is known that any priority matching is also a maximum-cardinality matching.