Given the powerset operator $\mathit P$, we have the following mapping
$\tag 1 \mathcal \Phi: \mathbb N \to \Phi(\mathbb N) $ $\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \, n \mapsto \mathit P^n(\mathbb N)$
What happens if we take away the Axiom of Power Set in $ZF\pm C$ and replace it with $\text{(1)}$? Would this contradict the other axioms?
I assume that by (1) you mean the existence of the operator $\Phi$ that you defined above. As you say, the existence of such a $\Phi$ is a consequence of the Powerset Axiom and of the other axioms of $ZF$, so if (1) contradicts the other axioms, the Powerset Axiom does as well.
Moreover, it seems to me that $V_{\omega_1}$ is a model of $ZFC-P+(1)$, so the consistency strength of this theory is strictly less that that of $ZFC$.