Solovay proved that a measurable cardinal is equiconsistent with the existence of an extension of Lebesgue measure to a full measure on $P(\mathbb R)$. The inner model direction is relatively simple. Let $\kappa$ be the additivity of the $\sigma$-ideal of measure zero sets, and translate this to a $\kappa$-complete ideal $J$ on $\kappa$. $P(\kappa)/J$ is a c.c.c. boolean algebra. If $F$ is the dual filter, then one can show using Kunen’s theory that $L[F]$ is a model of $\kappa$ measurable. The key is to show $L[F] \models$ GCH.

Are there similar equiconsistencies with larger cardinals, perhaps using non-canonical models? For example, suppose $\kappa$ is weakly inaccessible there is a $\kappa$-complete uniform ideal $I$ on $\kappa^+$ such that $P(\kappa^+)/I$ is c.c.c. Is $\kappa$ $\kappa^+$-strongly compact in an inner model? What is the best we can say about the consistency strength?

Update: Related easier question. Suppose $U$ is a $\kappa$-complete uniform ultrafilter on $\kappa^+$. Consider the model $L[U]$. Is $(\kappa^+)^{L[U]}<\kappa^+$?