We study soliton Thouless pumping in an extended diagonal Aubry-Andr'e-Harper model with on-site nonlinearities and inter-site nonlinearities. We show that the inter-site nonlinearities can make solitons acquire anomalous transport distances far beyond the ones predicted by the linear bands, and the quantized displacements can be engineered well. We uncover that nonlinear instabilities require lower limits on sweeping rates for soliton pumping, challenging the common notion that slower modulation enables a more favorable realization of topological transport. The nonlinear interactions between solitons make multi-soliton pumping generally lack the robustness characteristic of Thouless pumping as linear systems. Our results provide many possibilities to engineer topological pumping by nonlinearities, and further make a step for applications of soliton pumping.