My question essentially boils down to the following:
Given a countable transitive model (henceforth ctm) $M$ of ZFC-P (where P denotes the Power Set Axiom), $\mathbb{P}$ a forcing notion in $M$, and $G$ an $M$-generic filter, can we conclude that $M[G]\models$ ZFC-P?
Looking at Kunen's treatment of forcing, both the Truth and the Definability Lemmas are stated for ctm of ZF-P. However, when reading through the proof that the generic extension satisfies ZF-P, there is an appeal to the Reflection Theorem (within $M$) to show that each instance of Replacement holds in $M[G]$:
A similar thing happens in Schindler's Set Theory, where he uses Scott's Trick to cut down a collection of names to a set in $M$, thereby relying on Power Set. Can this be circumvented? I'm not sure whether this happens in the Boolean valued approach, but I would wager that constructing completions for arbitrary posets might be tricky without Power Set.
It would be nice if this was doable, because one can produce, in ZFC, countable transitive models of ZFC-P + "for every $x$, $[x]^{\aleph_0}$ exists", so then we could get actual set models of the full ZFC-P + "for every $x$, $[x]^{\aleph_0}$ exists" + ¬CH, via the usual forcing argument. In fact, the following is stated in Kanovei's Borel Equivalence Relations, lemma A.3.1:
If $\mathfrak{M}$ is a countable transitive model of ZFC - P + "for every $x$, $[x]^{\aleph_0}$ exists", $P$ is countable in $\mathfrak{M}$ and $G$ is $P$-generic over $\mathfrak{M}$, then $\mathfrak{M}[G]$ is still a model of ZFC-P+"for every $x$, $[x]^{\aleph_0}$ exists".
EDIT:
Something I thought of while working on completely unrelated stuff: If we have that the ground model $M$ satisfies Collection, instead of Replacement, then Kunen's proof goes through, without any appeal to the Reflection Theorem. So, if Collection holds in the ground model, then it holds in the extension. But producing ctm of ZFC-P with Collection instead of Replacement is easy, since $H_\kappa$ models this theory for regular uncountable $\kappa$.
Is the last paragraph correct? I don't immediately see a flaw.