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We analyze the effects of the long-range Coulomb interaction on the distribution of Berry curvature among the bands near charge neutrality of twisted bilayer graphene (TBG) closely aligned with hexagonal boron nitride (hBN). Due to the suppressed dispersion of the narrow bands, the band structure is strongly renormalized by electron-electron interactions, and thus, the associated topological properties of the bands are sensitive to filling. Using a Hartree formalism, we calculate the linear and nonlinear Hall conductivities, and find that for certain fillings, the remote bands contribute substantially to the Hall currents while the contribution from the central bands is suppressed. In particular, we find that these currents are generically substantial near regions of energies where the bands are highly entangled with each other, often featuring doping-induced band inversions. Our results demonstrate that topological transport in TBG/hBN is substantially modified by electron-electron interactions, which offer a simple explanation to recent experimental results.