This is just like the "metric of flat spacetime and EM potential"
but with the Newton near-field approximation added.
In fact, it's the metric listed halfway down thte "gravitomagnetics" page.

To match the gravitomagnetics page, I think I'll use +--- signature and not use natural units.

$\eta_{ab} = \eta^{ab} = \left[\matrix{ 1 & 0 \\ 0 & -\delta_{ij} }\right]$

I'm going to set $A_t = \frac{1}{c} \Phi^g$
I'm also going to scale down the $A_i$ vector by $\frac{1}{4}$

With all these changed signatures comes a changed definition of $F_{ab}$:
$F_{ab} = 2 {A^e}_{[b,a]} = \left[\matrix{ 0 & \frac{1}{c} E_j \\ -\frac{1}{c} E_i & -{\epsilon_{ij}}^k B_k }\right]$

$g_{ab} = \left[\matrix{ 1 + \frac{2}{c} A_t & -\frac{1}{c} A_x & -\frac{1}{c} A_y & -\frac{1}{c} A_z \\ -\frac{1}{c} A_x & -1 + \frac{2}{c} A_t & 0 & 0 \\ -\frac{1}{c} A_y & 0 & -1 + \frac{2}{c} A_t & 0 \\ -\frac{1}{c} A_z & 0 & 0 & -1 + \frac{2}{c} A_t }\right]$ $ = \left[\matrix{ 1 + \frac{2}{c} A_t & -\frac{1}{c} A_j \\ -\frac{1}{c} A_i & \delta_{ij} (-1 + \frac{2}{c} A_t) }\right]$ $ = \eta_{ab} + \frac{2}{c} A_t \delta_{ab} - \frac{1}{c} \delta_{t(b)} \hat{A}_{a)}$ For $\hat{A}_a = (0, A_i)$

Determinant:
$g = \frac{1}{c^4} ( -c^2 + 4 A_t^2 - A_x^2 - A_y^2 - A_z^2 ) (c + 2 A_t)^2 $