Well, here is a very slight weakening of your $\kappa^+$-supercompactness upper bound, to the assumption merely that $\kappa$ is nearly nearly $\kappa^+$-supercompact. This hypothesis strictly weaker than $\kappa^+$-supercompactness, but still under this assumption the set of such $a$ is stationary as you desire.

Specifically, a cardinal $\kappa$ is nearly $\kappa^+$-supercompact, if for every transitive $M$ of size $\kappa^+$ there is an elementary embedding $j:M\to N$ into a transitive set $N$, with critical point $\kappa$, such that $j''\kappa^+\in N$. It follows that $j''\kappa^+$ has your property with respect to $j(\kappa)$ in $N$, and so the set of $a\in P_\kappa(\kappa^+)$ with your property will have measure one with respect to the induced $M$-measure. It follows by the usual normality argument that the set of such $a$ is stationary, as you desire, since any club set $C$ (or the function defining it) can be placed into such an $M$ and so $j''\kappa^+\in j(C)$, meaning that $j$ of the set of $a$ meets $j(C)$, and so there is such an $a$ in $C$.

The nearly $\theta$-supercompact cardinals were introduced by Jason Schanker in his dissertation (written under my supervision), along with the weakly measurable cardinals, and Jason had noted that the near $\theta$-supercompactness hypothesis suffice in many arguments that previously had used full $\theta$-supercompactness. The present case is an additional instance of this phenomenon, where near $\kappa^+$-supercompactness suffices in place of $\kappa^+$-supercompactness.

Meanwhile, I am unsure exactly how the nearly $\kappa^+$-supercompact cardinals relate to the subcompact cardinals, and in any case perhaps a more dramatic weakening is possible.

It may be interesting to note that a recent result proved jointly by Jason Schanker, Brent Cody, Moti Gitik and myself shows that it is relatively consistent with ZFC that the least weakly compact cardinal $\kappa$ is also nearly $\kappa^+$-supercompact (but $2^\kappa\gt\kappa^+$), which might suggest something about the nature of these cardinals.

Well, here is a very slight weakening of your $\kappa^+$-supercompactness upper bound, to the assumption merely that $\kappa$ is   nearly $\kappa^+$-supercompact. This hypothesis is strictly weaker than $\kappa^+$-supercompactness, but still, under this assumption, the set of such $a$ is stationary as you desire.

Specifically, a cardinal $\kappa$ is nearly $\kappa^+$-supercompact, if for every transitive $M$ of size $\kappa^+$, closed under $\lt\kappa$ sequences, there is an elementary embedding $j:M\to N$ into a transitive set $N$, with critical point $\kappa$, such that $j''\kappa^+\in N$. It follows that $j''\kappa^+$ has your property with respect to $j(\kappa)$ in $N$, and so the set of $a\in P_\kappa(\kappa^+)$ with your property will have measure one with respect to the induced $M$-measure. It follows by the usual normality argument that the set of such $a$ is stationary, as you desire, since any club set $C$ (or the function defining it) can be placed into such an $M$ and so $j''\kappa^+\in j(C)$, meaning that $j$ of the set of $a$ meets $j(C)$, and so there is such an $a$ in $C$.

The nearly $\theta$-supercompact cardinals were introduced by Jason Schanker in his dissertation (written under my supervision), along with the weakly measurable cardinals, and Jason had noted that the near $\theta$-supercompactness hypothesis suffice in many arguments that previously had used full $\theta$-supercompactness. The present case is an additional instance of this phenomenon, where near $\kappa^+$-supercompactness suffices in place of $\kappa^+$-supercompactness.

Meanwhile, I am unsure exactly how the nearly $\kappa^+$-supercompact cardinals relate to the subcompact cardinals, and in any case perhaps a more dramatic weakening is possible.

It may be interesting to note that a recent result proved jointly by Jason Schanker, Brent Cody, Moti Gitik and myself shows that it is relatively consistent with ZFC that the least weakly compact cardinal $\kappa$ is also nearly $\kappa^+$-supercompact (but $2^\kappa\gt\kappa^+$), which might suggest something about the nature of these cardinals.