A cardinal $\kappa$ is regular if (and only if) the union of fewer than $\kappa$ many sets of size less than $\kappa$ still always has size less than $\kappa$. That seems to be exactly what you have here. Also, the union of downward closed sets remains downward closed, so you don't need to take the downward closure of the union, as it is already downward closed.

A cardinal $\kappa$ is regular if (and only if) the union of fewer than $\kappa$ many sets of size less than $\kappa$ still always has size less than $\kappa$. That seems to be exactly what you have here. Also, the union of downward closed sets remains downward closed, so you don't need to take the downward closure of the union, as it is already downward closed.

Note, however, that the downward closure of a $\kappa$-small family might no longer be $\kappa$-small, if $P$ has large initial segments. For example, $P$ may have no $\kappa$-small downward closed subposets at all (this is true in the reverse ordinal $\kappa^*$, which is $\kappa$ turned upside down).