  Upcoming cosmic microwave background (CMB) lensing measurements and
tomographic galaxy surveys are expected to provide us with high-precision data
sets in the coming years, thus paving the way for fruitful cross-correlation
analyses. In this paper we study the information content of the weighted
skew-spectrum, a nearly-optimal estimator of the angular bispectrum amplitude,
as a means to extract non-Gaussian information on both bias and cosmological
parameters from the bispectra of galaxies cross-correlated with CMB lensing,
while gaining significantly on speed. Our results show that for the combination
of the Planck satellite and the Dark Energy Spectroscopic Instrument (DESI),
the difference in the constraints on bias and cosmological parameters from the
skew-spectrum and the bispectrum is at most $17\%$. We further compare and find
agreement between our theoretical skew-spectra and those estimated from N-body
simulations, for which it is important to include gravitational non-linearities
beyond perturbation theory and the post-Born effect for CMB lensing. We define
an algorithm to apply the skew-spectrum estimator to the data and, as a
preliminary step, we use the skew-spectra to constrain bias parameters and the
amplitude of shot noise from the simulations through a Markov chain Monte Carlo
likelihood analysis, finding that it may be possible to reach percent-level
estimates for the linear bias parameter $b_1$.
