  Developing analysis pipelines based on statistics beyond two-point functions
is critical for extracting a maximal amount of cosmological information from
current and upcoming weak lensing surveys. In this paper, we study the impact
of the intrinsic alignment of galaxies (IA) on three promising probes measured
from aperture mass maps -- the lensing peaks, minima and full PDF, in
comparison and in combination with the shear two-point correlation functions
($\gamma$-2PCFs). Our two-dimensional IA infusion method converts the
light-cone-projected mass sheets into projected tidal tensors, which are then
linearly coupled to an intrinsic ellipticity component with a strength
controlled by the coupling parameter $A_{\rm IA}$. We validate our method with
the $\gamma$-2PCFs statistics, recovering well the analytical calculations from
the linear alignment model of Bridle \& King in a full tomographic setting, and
for different $A_{\rm IA}$ values. We next use our method to infuse at the
galaxy catalogue level a non-linear IA model that includes the
density-weighting term introduced in \citet{Blazek2015}, and compute the impact
on the three aperture mass map statistics. We find that large \snr peaks are
maximally affected, with deviations reaching 30\% (10\%) for a {\it
Euclid}-like (KiDS-like) survey. Modelling the signal in a $w$CDM cosmology
universe with $N$-body simulations, we forecast the cosmological bias caused by
unmodelled IA for 100 deg$^2$ of {\it Euclid}-like data, finding very large
offsets in $w_0$ (5-10$\sigma_{\rm stat}$), $\Omega_{\rm m}$ (4-6$\sigma_{\rm
stat}$), and $S_8 \equiv \sigma_8\sqrt{\Omega_{\rm m}/0.3}$
($\sim$3$\sigma_{\rm stat}$). The method presented in this paper offers a
compelling avenue to account for IA in beyond-two-point weak lensing
statistics, with a flexibility comparable to that of current $\gamma$-2PCFs IA
analytical models.
