= Solution
Here $\overline u$ is the <Dirac field>'s adjoint, not a second independently defined charge-conjugate field. <Charge conjugation of a Dirac field> gives $u\mapsto C\overline u^T$ and $\overline u\mapsto-u^T C^{-1}$; <parity> gives $u(x)\mapsto\gamma^0u(x_P)$ and $\overline u(x)\mapsto\overline u(x_P)\gamma^0$. With the product convention $\widehat{CP}=\widehat C\widehat P$, therefore,
$$
\boxed{u(x)\longmapsto\gamma^0C\overline u(x_P)^T,\qquad
\overline u(x)\longmapsto-u(x_P)^T C^{-1}\gamma^0.}
$$
Reversing the convention for the operator product changes an overall intrinsic phase, which does not change the bilinear transformations when used consistently. The transpose also exchanges the fundamental colour index with its conjugate.
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