We investigate late–time cosmology in the Finsler–Barthel–Kropina framework, where anisotropic effects are introduced into the FLRW background through a time–dependent function $\eta (t)$ constructed via the osculating Riemannian approach. The resulting modified Friedmann equations generate direction–dependent corrections to the expansion history. Using Cosmic Chronometers (CC), DESI BAO, and Pantheon+ supernova observations, we reconstruct the Hubble rate, energy density, and key cosmographic quantities. The model yields $H_{0}$ values consistent with late–time observations and negative value of deceleration parameter at present indicates ongoing acceleration phase of universe, while the reconstructed jerk and $Om(z)$ diagnostics show clear departures from constant– $\Lambda $ evolution, indicating effective dark-energy like dynamics sourced by Finslerian anisotropy. The predicted cosmic age is slightly higher than $\Lambda $ CDM. A model selection analysis based on information criteria shows that the Finsler–Kropina model performs competitively with $\Lambda $ CDM for the CC+ DESI BAO dataset, whereas the full joint dataset mildly favors $\Lambda $ CDM due to its lower parameter complexity. This demonstrates that Finsler–Kropina geometry offers a viable anisotropic extension of late–time cosmology.
