The Axiom of Determinacy (AD) implies that the club filter on $\omega_1$ (the subsets of $\omega_1$ containing a club) is an ultrafilter. Certainly if that is the case then we can't even decompose $\omega_1$ into two disjoint stationary sets, because one of them would have to contain a club. Assuming sufficient large cardinal hypotheses (infinitely many Woodin cardinals and a measurable cardinal above them) one has that $L(\mathbb{R})$ satisfies AD, and hence that is a canonical inner model of the form I think you are looking for.

Of course

What about above $\omega_1$? I believe it is a theorem of John Steel's that (again under large cardinal assumptions) in $L(\mathbb{R})$ for any regular $\kappa$ below $\Theta$, the $\omega$-club filter on $\kappa$ is an ultrafilter. (An $\omega$-club is an unbounded set closed under countable limits). So for such $\kappa$ the stationary set of ordinals of countable cofinality cannot be partitioned into two disjoint stationary sets. I don't know about getting Solovay's theorem to fail at cardinals higher than that.

Also, that all assumes some large cardinals. I do not know if large cardinal assumptions are necessary to get the failure of Solovay's Theorem.

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The Axiom of Determinacy (AD) implies that the club filter on $\omega_1$ (the subsets of $\omega_1$ containing a club) is an ultrafilter. Certainly if that is the case then we can't even decompose $\omega_1$ into two disjoint stationary sets, because one of them would have to contain a club. Assuming sufficient large cardinal hypotheses (infinitely many Woodin cardinals and a measurable cardinal above them) one has that $L(\mathbb{R})$ satisfies AD, and hence that is a canonical inner model of the form I think you are looking for.

Of course that assumes some large cardinals. I do not know if large cardinal assumptions are necessary to get the failure of Solovay's Theorem.