This may be an easy question. Are the cardinals below the critical point of a precipitous ideal embedding always absolute between the generic ultrapower and the generic extension?

To focus on the simplest case, suppose $I$ is a precipitous normal ideal on $\omega_2$. Let $G$ be generic for $\mathcal{P}(\omega_2)/I$. Let $j : V \to M$ be the generic ultrapower. Then $\omega_1^V$ is not moved by $j$, but could it be the case that $\omega_1^V$ is countable in $V[G]$? Of course the answer is no if $I$ has enough presaturation.

**Edit:** As a subproblem towards a consistent negative answer, can we find (from some large cardinal assumption) a measurable $\kappa$ with measure $U$, and a $\kappa$-c.c. forcing $\mathbb{P}$ such that $\mathbb{P}$ preserves $\omega_1$, but $j_U(\mathbb{P})$ collapses $\omega_1$? (Not in $Ult(V,U)$ of course, but in $V$.)