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$$(\psi_{a}^{\ast} \gamma^0_{ab} \chi_{b})^{\ast} = \psi_{a} \gamma^{0\ast}_{ab} \chi_{b}^{\ast}$$ is not correct. Rather $$(\psi_{a}^{\ast} \gamma^0_{ab} \chi_{b})^{\ast} =\chi_{b}^{\ast} \gamma^{0\ast}_{ab} \psi_{a}= - \psi_{a} \gamma^{0\ast}_{ab} \chi_{b}^{\ast}$$ Note that the Hermitian is defined as $$(AB)^\dagger = B^\dagger A^\dagger$$ Note ...