We study the spin-1 Babujian-Takhtajan chain deformed by its third conserved charge Q₃. We derive Q₃ and show that it is a dimensionless energy current and that its local density is a dressed scalar-chirality operator rather than bare chirality alone, as is the case for the spin-1/2 Heisenberg chain. The deformation H_α=H+αQ₃ therefore provides a local, exactly solvable current bias: it leaves the eigenstates of the original Hamiltonian unchanged, but reorders them so that selected high-energy current-carrying states become ground states of the tilted problem. Using the thermodynamic Bethe ansatz and confirming the analytical calculations with DMRG, we find a quantum phase transition at α_c=J/(8π). For α<α_c, the ground-state remains the undeformed Babujian-Takhtajan phase whose low-energy effective field theory is described by the SU(2) Wess-Zumino-Witten (WZW) model at level k=2 representing a critical phase characterized by a central charge c=3/2 and ⟨ Q₃⟩=0. For α>α_c, a finite rapidity interval forms, and the system enters a gapless chiral current-carrying sector described by a c=3/2 CFT. Near the threshold, the free energy starts quadratically as a function of α-α_c, while the energy current turn on linearly. The scalar chirality turns on at the same threshold, showing that the postcritical sector is simultaneously current-carrying and chiral. The most immediate experimental routes are composite spin-1 bosons in optical lattices, and programmable qutrit simulators based on trapped ions or superconducting circuits.