CPT theorem and the weak interaction

I am trying the understand how the weak interaction obeys CPT symmetry. My understanding is that, under a CPT transformation, the Lagrangian for the Standard Model should be invariant, and thus the term that dictates the interaction between quarks and the W boson,

$$L = frac{g}{2sqrt{2}}(J^{mu}W^{+}_{mu}+ J^{mu}W^{-}_{mu}),$$
$$J^mu=bar{u}^i_Lgamma_{mu}V^{ij}_{CKM}d_L^i$$
should be invariant under conjugation by $Theta=hat{C}hat{P}hat{T}.$

However, it is difficult to see that this will invariance will hold using the usual spinor transformations

$$hat{P}psihat{P}^{-1}=gamma^0psi$$
$$hat{C}psihat{C}^{-1}=Cbarpsi^T$$
$$hat{T}psihat{T}^{-1}=Bpsi.$$

Is it possible to show CPT symmetry using these transformations? If not, why not, and how does CPT symmetry manifest itself otherwise?