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. Our 2D 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_IA. We validate our method with the γ-2PCFs statistics, recovering well the linear alignment model of Bridle & King in a full tomographic setting, and for different A_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 Blazek et al., and compute the impact on the three aperture mass map statistics. We find that large S/N peaks are maximally affected, with deviations reaching 30 per cent (10 per cent) for a Euclid-like (KiDS-like) survey. Modelling the signal in a wCDM cosmology universe with N-body simulations, we forecast the cosmological bias caused by unmodelled IA for 100 deg^2 of Euclid-like data, finding very large offsets in w_0 (5-10σ_stat), Ω_m (4-6σ_stat), and S8≡σ8√Ωm/0.3 (∼3σ_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 γ-2PCFs IA analytical models.