I once derived something similar starting from the momentum equation of a multi-fluid model:
\begin{equation}
m_{\alpha}n_{\alpha}\left[\partial_{t}\mathbf{v}_{\alpha}+\left(\mathbf{v}_{\alpha}\cdot\nabla\right)\mathbf{v}_{\alpha}\right]=n_{\alpha}q_{\alpha}\left(\mathbf{E}+\mathbf{v}_{\alpha}\times\mathbf{B}\right)-\nabla\cdot\boldsymbol{P}_{\alpha}+\sum_{\beta}\mathbf{R}_{\alpha\beta}\label{eq:2fluid_1}
\end{equation}
Where $m$ and $q$ denote mass and charge of the fluid elements respectively.
This equation decribes the conservation of momentum for each particle
species $\alpha$ (i.e. Newton's 2nd law). On the left hand
side we have inertial forces due to temporal or convective changes.
On the right hand side we have forces acting on the fluids. From left
to right these are the Lorentz force, internal forces described by
the pressure tensor (pressure or viscosity) and a term which allows
the momentum transfer from species $\alpha$ to species $\beta$.
Depending on your exact situation you can now cross terms out if they are not that important, like in your case the pressure. Since I am working on magnetic confinement in fusion devices it was save for me to neglect the whole left side since it is way smaller in these conditions than the vxB term on the right. Depending on your situation you may do the same.
If we assume that the plasma is made of electrons and one species of
ions, the term respecting transfer of momentum to electrons by electron-ion
collisions can be written as
\begin{equation}
\mathbf{R}_{ei}=-m_{e}n_{e}\nu_{ei}\left(\mathbf{v}_{e}-\mathbf{v}_{i}\right),
\end{equation}
where $\nu_{ei} = \tau_{ei}^{-1}$ is the collisionality or inverse collision time. This is called friction force and already looks similar to a current $\mathbf{J}=-n_{e}e\left(\mathbf{v}_{e}-\mathbf{v}_{i}\right)$. With only electrons and ions you will have 2 version of the momentum equation above, one for each species. This is usually called 2-fluid model.
I think you may use this as a starting point since there is now a current and the electric field in the equations and the combination of constants will make up a conductivity
\begin{equation}
\sigma =\frac{n_{e}e^{2}}{m_{e}\nu_{ei}},
\end{equation}
in the end.
