  The wavelet scattering transform (WST) has recently gained attention in the
context of large-scale structure studies, being a possible generator of summary
statistics encapsulating non-Gaussianities beyond the reach of the conventional
power spectrum. This work examines the three-dimensional solid harmonic WST in
the context of a three-dimensional line-intensity mapping measurement to be
undertaken by current and proposed phases of the CO Mapping Array Project
(COMAP). The WST coefficients demonstrate interpretable behaviour in the
context of noiseless CO line-intensity simulations. The contribution of the
cosmological $z\sim3$ signal to these coefficients is also detectable in
principle even in the Pathfinder phase of COMAP. Using the peak-patch method to
generate large numbers of simulations and incorporating observational noise, we
numerically estimate covariance matrices and show that careful choices of WST
hyperparameters and rescaled or reduced coefficient sets are both necessary to
keep covariances well-conditioned. Fisher forecasts show that even a reduced
`shapeless' set of $\ell$-averaged WST coefficients show constraining power
that can exceed that of the power spectrum alone even with similar detection
significance. The full WST could improve parameter constraints even over the
combination of the power spectrum and the voxel intensity distribution, showing
that it uniquely encapsulates shape information about the line-intensity field.
However, practical applications urgently require further understanding of the
WST in key contexts like covariances and cross-correlations.
