Here's a very silly mistake I made recently: I claimed that if $\mathbb{P}\in L(\mathbb{R})$ is a forcing which adds a real, then $$(*)\quad L(\mathbb{R})^{V^\mathbb{P}}=L(\mathbb{R})^\mathbb{P}.$$ While it is true (**EDIT: no it isn't, see the answer below**) that $L(\mathbb{R})^\mathbb{P}=L(\mathbb{R})^{(L(\mathbb{R})^\mathbb{P})}$, what I wrote above is nonsense: $V$ may have names for reals which are not in $\mathbb{R}$.

Indeed, from large cardinals we can prove that $(*)$ *fails* - if it held, then examining $Col(\omega,\kappa)$ for $\kappa$ regular in $V$, we'd have that $L(\mathbb{R})$ contains a proper class of measurables; and this is known to not be the case, assuming large cardinals (specifically, from large cardinals there are no measurable cardinals in $L(\mathbb{R})$ above $\Theta$).

My question is around principles of the form $(*)$. Specifically, for a definable class of forcings $\mathcal{C}$ which add a real, let (FC)$_\mathcal{C}$ ("forcing conflation") be the statement $(*)$ restricted to forcings in $\mathcal{C}\cap L(\mathbb{R})$. I'm interested in when (FC)$_\mathcal{C}$ is compatible with large cardinals - specifically, a proper class of Woodins - and when it adds large cardinal strength to ZFC+proper class of Woodins.

To make this concrete, I'll ask:

Is (FC)$_{proper}$ consistent with ZFC + "There is a proper class of Woodins," relative perhaps to even stronger large cardinal hypotheses? If so, does ZFC + "There is a proper class of Woodins" + (FC)$_{proper}$ have consistency strength beyond a proper class of Woodins?