The interplay between crystal structure and electronic band structure has recently encouraged the study of magnetic materials that exhibit intriguing anomalous Hall effect (AHE). In this context, the magnetic shape memory (MSM) alloy undergoing structural phase transition is an ideal platform for investigating its effect on transport phenomena. Here, we report various aspects of the scattering mechanism in the Mn${}_2$NiGa MSM Heusler alloy. In the low-symmetry martensite phase, the scaling behavior of anomalous Hall resistivity reflects that both extrinsic skew scattering and intrinsic mechanism have nearly equal contributions to the AHE. The obtained value of experimental intrinsic anomalous Hall conductivity (IAHC) is 12 S/cm, an order of magnitude lower than the calculated theoretical IAHC, possibly due to twin domain formation or modulated structures in the martensite phase. In contrast, in the high-symmetry austenite phase, the scaling behavior reveals the dominating extrinsic contribution compared to the intrinsic one. Notably, both the quadratic temperature dependence of resistivity and the linear negative magnetoresistance confirm the dominance of electron-magnon coupling in the Mn${}_2$NiGa alloy. This coupling plays a crucial role by driving the side-jump scattering mechanism, which significantly influences the AHE in the austenite phase. The experimental results suggest that the intrinsic mechanism is being suppressed during transition from low-symmetry martensite to high-symmetry austenite phase. This is further qualitatively supported by theoretical IAHC calculations, which indicate a lower IAHC value in the austenite phase compared to the martensite phase. These findings indicate that the intrinsic contribution to the AHE weakens as the alloy transforms to the high-symmetry austenite phase, where the strong electron-magnon coupling emerges as a key factor in shaping the transport behavior of the Mn${}_2$NiGa MSM alloy.