We report on a temperature-driven reversible change of the in-plane magnetic anisotropy of V$_2$O$_3$/Ni bilayers. This is caused by the rhombohedral to monoclinic structural phase transition of V$_2$O$_3$ at $T_C$ = 160 K. The in-plane magnetic anisotropy is uniaxial above $T_C$, but as the bilayer is cooled through the structural phase transition, a secondary magnetic easy axis emerges. Ferromagnetic resonance measurements show that this change in magnetic anisotropy is reversible with temperature. We identify two structural properties of the V$_2$O$_3$/Ni bilayers affecting the in-plane magnetic anisotropy: (1) a growth-induced uniaxial magnetic anisotropy associated with step-like terraces in the bilayer microstructure and (2) a low-temperature strain-induced biaxial anisotropy associated with the V$_2$O$_3$ structural phase transition. Magnetoresistance measurements corroborate the change in magnetic anisotropy across the structural transition and suggest that the negative magnetostriction of Ni leads to the emergence of a strain-induced easy-axis. This shows that a temperature-dependent structural transition in V$_2$O$_3$ may be used to tune the magnetic anisotropy in an adjacent ferromagnetic thin film.