Choice principles in local mantles

Assume ZFC. Let $\kappa$ be a cardinal. A $<\kappa$-ground is a transitive proper class $W$ modelling ZFC and such that $V$ is a generic extension of $W$ via a forcing $\mathbb{P}\in W$ of cardinality $<\kappa$. The $\kappa$-mantle is the intersection of all $<\kappa$-grounds. We prove that certain partial choice principles in the $\kappa$-mantle are the consequence of $\kappa$ being inaccessible/weakly compact, and some other related facts.