Suppose that $P \subset M$ is the given two-sided real projective plane. Let $M'$ be the orientation double cover of $M$. Let $S$ be the two-sphere in $M'$ that double covers $P$.

Exercise: $P$ is non-separating in $M$ if and only if $S$ is non-separating in $M'$.

Exercise: If $S$ separates $M'$, then neither component of $M' - S$ is a three-ball.

So in either case, $\pi_2(M')$ is non-trivial; thus $\pi_2(M)$ is non-trivial.

Suppose that $P \subset M$ is a two-sided real projective plane. Let $M'$ be the orientation double cover of $M$. Let $S$ be the two-sphere in $M'$ that double covers $P$.

Exercise A: $P$ is non-separating in $M$ if and only if $S$ is non-separating in $M'$.

Exercise B: If $S$ separates $M'$, then neither component of $M' - S$ is a three-ball.

We deduce, in both the separating and non-separating cases, that $\pi_2(M')$ is non-trivial; thus $\pi_2(M)$ is non-trivial.