  The predicted present-day amplitude of matter fluctuations based on cosmic
microwave background (CMB) anisotropy data has sometimes been found discrepant
with more direct measurements of late-time structure. This has motivated many
extensions to the standard cosmological model, including kinetic interactions
between dark matter and dark energy that introduce a drag force slowing the
growth of structure at late times. Exploring this scenario, we develop a model
for quasi-linear scales in the matter power spectrum by calculating the
critical overdensity in the presence of this interaction and a varying dark
energy equation of state. We explicitly avoid modeling or interpretation of
data on non-linear scales in this model (such as use of $\Lambda$CDM-calibrated
priors), which would require numerical simulations. We find that the presence
of the drag force hinders halo formation, thus increasing the deviation from
$\Lambda$CDM in the mildly non-linear regime. We use CMB lensing observations
from the sixth data release of the Atacama Cosmology Telescope up to $L=1250$
(in combination with Planck, Sloan Digital Sky Survey, and 6dFGS data) to
derive the strongest constraints to date on the amplitude of the drag term,
finding the dimensionless interaction strength
$\Gamma_\mathrm{DMDE}/(H_0\rho_\mathrm{c})<0.831\; (2.81)$ at the 68\% (95\%)
confidence level. The inclusion of non-linear corrections improves our
constraints by about 35\% compared to linear theory. Our results do not exclude
the best-fit values of $\Gamma_\mathrm{DMDE}$ found in previous studies using
information from galaxy weak lensing, though we find no statistical preference
for the dark matter-dark energy kinetic interactions over $\Lambda$CDM. We
implement our model in a publicly available fork of the Boltzmann code CLASS at
https://github.com/fmccarthy/Class_DMDE.
