Does this problem map to the Set Packing problem?
You cannot reduce set-packing to your problem, unless P=NP.

There is a linear-time algorithm for your problem.

Remove all vertices of type 1 and all their neighbors. Let $m'$  (resp. $n'$) be the set of remaining vertices in $m$ (resp. $n$). 
If $|m'| \ge k$ and the neighborhood of $m$ contains all the vertices in $n'$, the answer is yes.

If any of the above two conditions is not satisfied then the answer is no:


If $|m'| &lt; k$ then any set $S \subseteq m$ with $|S| \ge k$ contains a vertex that is adjacent to some vertex of type 1 in $n$.
If there is a vertex $v \in n$ of type 2 that is not in the neighborhood of $m'$, then every vertex $u \in m$ that is adjacent to $v$ is also adjacent to a vertex of type 1 in $n$.