It is known that if $\delta$ is a Woodin cardinal and $\kappa < \delta$, then the stationary tower forcing $\mathbb Q^\kappa_{<\delta}$ adds no bounded subsets of $\kappa$ and forces $\delta = \kappa^+$. Thus if there is a Woodin cardinal $\delta$ then there is a forcing adding no bounded subsets of $\aleph_\omega$ and making $\delta = \aleph_{\omega+1}$. But it is also known that $\mathbb Q^\kappa_{<\delta}$ is not $\delta$-c.c.

Question: Is there some large cardinal assumption that implies the existence of a cardinal $\kappa$ and a $\kappa$-c.c. forcing $\mathbb P$ which adds no bounded subsets of $\aleph_\omega$ and makes $\kappa = \aleph_{\omega+1}$?

I'm no expert on these things, but naively I would suggest two possible approaches: (a) Find a large cardinal $\delta$ that implies the existence of a $\delta$-saturated tower of ideals with similar effects as the stationary tower. (b) Find an inaccessible cardinal $\delta$ with a precipitous tower of ideals of height $\delta$ that adds no bounded subsets of $\aleph_\omega$ but actually collapses $\delta$, so that $\delta^+$ is the witness.