We study nematodynamics of a mesoscopic system consisting of sheared biaxial liquid crystalline polymers using a hydrodynamical kinetic theory, in which the biaxial liquid crystal polymer is modeled as a rigid, biaxial, ellipsoidal molecule immersed in viscous solvent. The governing Smoluchowski equation in the model is solved in selected regions of the material parameter space and a range of accessible shear rates using a Wigner-Galerkin spectral method. In addition to the truly biaxial flow-aligning steady states, log-rolling steady states, and out-of-plane steady states, we report the presence of two new time-periodic motions, chaotic motion and associated phase transitions in a range of shear rates and selected material parameters. Rheological signatures of the sheared mesoscopic system are identified with predominant shear thinning in all phases and alternating signs between the normal stress differences in steady vs time-dependent motions.