We report a theoretical study of the spin-dependent transport properties in heterostructures containing a three-dimensional topological insulator (TI) thin film and ferromagnetic normal insulator (FMNI) slab. Within the framework of a continual approach for the FMNI/TI/FMNI trilayer model, we reveal how the magnetic proximity effect at the TI/FMNI interface can influence an intrinsic Hall response of the system. We predict that the FMNI/TI/FMNI trilayer undergoes a transition into the quantum anomalous Hall phase either from the topologically trivial phase or from the quantum spin Hall phase, which is controlled by tuning the proximity-induced exchange field, the TI film thickness, and the TI band structure parameters. We draw the corresponding phase diagram of the FMNI/TI/FMNI trilayer. Moreover, we argue that the roughness at the TI/FMNI interfaces can cause the decomposition of the TI film into topologically distinct domains, which affects the Hall conductivity. We discuss the specifics of manifestation and complexities of observation of quantized conductivity in realistic TI/FMNI heterostructures.