  We report updated results for $B_K$ using HYP-smeared staggered valence
quarks on MILC asqtad lattices based on an analysis using SU(3) staggered
chiral perturbation theory. The most important new feature of our data sample
is the inclusion of a fourth ("ultrafine") lattice spacing. This improves the
control over the continuum extrapolation and errors due our use of one-loop
perturbative matching. We present a complete updated error budget, which leads
to $B_K(\text{NDR}, \mu = 2 \text{GeV}) = 0.5309 \pm 0.0051 \pm 0.0424$ and
$\hat{B}_K = B_K(\text{RGI}) = 0.727 \pm 0.07 \pm 0.058$. The results of the
SU(3) analysis are inferior to those based on SU(2) staggered chiral
perturbation theory, primarily because of the dependence on the Bayesian priors
we use in the SU(3) fits.
