  With the advent of a new generation of cosmological experiments that will
provide high-precision measurements of the cosmic microwave background (CMB)
and galaxies in the large-scale structure, it is pertinent to examine the
potential of performing a joint analysis of multiple cosmological probes. In
this paper, we study the cosmological information content contained in the
one-loop power spectra and tree bispectra of galaxies cross-correlated with CMB
lensing. We use the FFTLog method to compute angular correlations in spherical
harmonic space, applicable for wide angles that can be accessed by forthcoming
galaxy surveys. We find that adding the bispectra and cross-correlations with
CMB lensing offers a significant improvement in parameter constraints,
including those on the total neutrino mass, $M_\nu$, and local non-Gaussianity
amplitude, $f_{\rm NL}$. In particular, our results suggest that the
combination of the Vera C. Rubin Observatory's Legacy Survey of Space and Time
(LSST) and CMB-S4 will be able to achieve $\sigma(M_\nu)=42$ meV from galaxy
and CMB lensing correlations, and $\sigma(M_\nu)=12$ meV when further combined
with the CMB temperature and polarization data, without any prior on the
optical depth.
