The non-Newtonian rheology of a dilute suspension of hydrodynamically interacting colloids is studied theoretically via active, nonlinear microrheology. In this Stokes-flow regime, a Brownian probe is driven through the suspension by a fixed external force; its motion distorts the configuration of background bath particles, which in turn alters probe motion. This interplay was utilized in our recent article to obtain the normal elements of the suspension stress via the nonequilibrium statistical mechanics theory [Chu and Zia, J. Rheol. 60(4), 755-781 (2016)]. In the present article, we focus on the normal stress differences N-1 and N-2, osmotic pressure Pi, and their evolution with the strength of the probe force and strength of hydrodynamic interactions. As hydrodynamic interactions grow from weak to strong, the influence of couplings between the stress and the entrained motion on N-1 changes with the strength of flow. When flow is strong, hydrodynamic interactions suppress the magnitude of N-1, owing to collision shielding that preserves structural symmetry. In contrast, when flow is weak, hydrodynamic interactions enhance disparity in normal stresses and, in turn, increase the magnitude of N-1. The first normal stress difference changes sign as flow strength increases from weak to strong, due strictly to the influence of elastic interparticle forces. Regardless of the strength of flow and hydrodynamic interactions, the second normal stress difference is identically zero owing to the axisymmetry of the microstructure around the probe. Hydrodynamic forces act to suppress the osmotic pressure for any strength of flow; when the forcing is strong, this effect is qualitative, reducing the flow-strength dependence from linear to sublinear as hydrodynamic interactions grow from weak to strong. Non-Newtonian rheology persists as long as entropic forces play a role, i.e., in the presence of particle roughness or even very weak Brownian motion, but vanishes entirely in the pure-hydrodynamic limit. (C) 2017 The Society of Rheology.