The relation $\:\mathbf f \boldsymbol{=} m_0 \mathbf a\:$ is not valid. Instead of it use this \begin{equation} \mathbf f \boldsymbol{=}\gamma_u m_0 \mathbf a\boldsymbol{+}\gamma^3_u m_0 \dfrac{\left(\mathbf a \boldsymbol{\cdot}\mathbf u\right)}{c^2}\mathbf u \quad \boldsymbol{\Longrightarrow} \quad \boxed{\:\:\mathbf f\boldsymbol{\cdot}\mathbf u \boldsymbol{=}\gamma^3_u m_0 \left(\mathbf a \boldsymbol{\cdot}\mathbf u\right)\vphantom{\dfrac{a}{b}}\:\:} \tag{A-01}\label{A-01} \end{equation}

The relation $\:\mathbf f \boldsymbol{=} m_0 \mathbf a\:$ is not valid. Instead of it use this \begin{equation} \mathbf f \boldsymbol{=}\gamma_u m_0 \mathbf a\boldsymbol{+}\gamma^3_u m_0 \dfrac{\left(\mathbf a \boldsymbol{\cdot}\mathbf u\right)}{c^2}\mathbf u \quad \boldsymbol{\Longrightarrow} \quad \boxed{\:\:\mathbf f\boldsymbol{\cdot}\mathbf u \boldsymbol{=}\gamma^3_u m_0 \left(\mathbf a \boldsymbol{\cdot}\mathbf u\right)\vphantom{\dfrac{a}{b}}\:\:} \tag{A-01}\label{A-01} \end{equation} To reach that combine \begin{equation} \mathbf f \boldsymbol{=}\dfrac{\mathrm d\mathbf p}{\mathrm d t} \boldsymbol{=}\dfrac{\mathrm d\left(\gamma_u m_0 \mathbf u\right)}{\mathrm d t} \boldsymbol{=}\cdots \tag{A-02}\label{A-02} \end{equation} with yours \begin{equation} \dfrac{\mathrm d\gamma_u}{\mathrm d t} \boldsymbol{=}\cdots \tag{A-03}\label{A-03} \end{equation}