Bulk directional enhancement factors are determined for axisymmetric (girdle and single-maximum) orientation fabrics using a transversely isotropic grain rheology with an orientation-dependent non-linear grain fluidity. Compared to grain fluidities that are simplified as orientation independent, we find that bulk strain-rate enhancements for intermediate-to-strong axisymmetric fabrics can be up to a factor of ten larger, assuming stress homogenization over the polycrystal scale. Our work thus extends previous results based on simple basal slip (Schmid) grain rheologies to the transversely isotropic rheology, which has implications for large-scale anisotropic ice-flow modelling that relies on a transversely isotropic grain rheology. In order to derive bulk enhancement factors for arbitrary evolving fabrics, we expand the c-axis distribution in terms of a spherical harmonic series, which allows the rheology-required structure tensors through order eight to easily be calculated and provides an alternative to current structure-tensor-based modelling.