We study the scaling properties of heterogeneities in nematic (liquid crystal) polymers that are generated by pressure-driven, capillary Poiseuille flow. These studies complement our earlier drag-driven structure simulations and analyses. We use the mesoscopic Doi-Marrucci-Greco model, which incorporates excluded-volume interactions of the rod-like particle ensemble, distortional elasticity of the dispersion, and hydrodynamic feedback through orientation dependent viscoelastic stresses. The geometry likewise introduces anchoring conditions on the nano-rods which touch the solid boundaries. We first derive flow-orientation steady-state structures for three different anchoring conditions, by asymptotic analysis in the limit of weak pressure gradient. These closed-form expressions yield scaling laws, which predict how lengthscales of distortions in the flow and orientational distributions vary with strength of the excluded volume potential, molecule geometry, and distortional elasticity constants. Next, the asymptotic structures are verified by direct numerical simulations, which provide a high level benchmark on the numerical code and algorithm. Finally, we calculate the effective (thermal or electrical) conductivity tensor of the composite films, and determine scaling behavior of the effective property enhancements generated by capillary Poiseuille flow.