Incorporating Gravity in special relativity and in Newtonian limit Consider the following metric:
$$ds^{2} = -c^{2}dt^{2} + dx^{2}+dy^{2} + dz^{2}. \tag{1}$$
This is the Minkowski Metric which describes a spacetime without Gravitational interaction. Furthermore, this is the basic background spacetime of Quantum Field Theory. 
Now, suppose we introduce Gravity in Special Relativity, then we are lead to Equivalence Principle and the more "close to Special Relativity" geometry is in fact a General Relativistic approximation expressed on:
$$ ds^{2} = -\Big(1+\frac{2\Phi(x',y',z')}{c^{2}}\Big)c^{2}dt^{2}+\Big(1-\frac{2\Phi(x',y',z')}{c^{2}}\Big)(dx'^{2}+dy'^{2} + dz'^{2}).\tag{2}$$
But suppose we do not have General Relativity yet, just Special Relativity and Equivalence Principle. Due to gravitational redshift effects can we introduce a metric as,
$$ ds^{2} = -\Big(1+\frac{2\Phi(x',y',z')}{c^{2}}\Big)c^{2}dt^{2}+(dx'^{2}+dy'^{2} + dz'^{2}).\tag{3}$$
to incorporate this gravitational redshift and therefore, a candidate to gravitational effects in SR?
$$* * *$$
I already asked a similar question, but my point of view as totally wrong: Doubt on Newtonian weak field metric, accelerated frames and metric tensor transformation
 A: You can introduce such a metric, and in fact it reproduces the effects of Newtonian physics to a pretty good accuracy.  Specifically, the geodesic equation for this metric becomes
$$
\frac{d^2t}{d\tau^2} = 0 \\
\frac{d^2 \vec{x}}{d \tau^2} = - \vec{\nabla} \Phi \left( \frac{dt}{d\tau} \right)^2
$$
in units where $c = 1$.  The first equation effectively just tells us that we can set our units and origin for the time coordinate as we please;  an obvious choice is simply $t = \tau$.   This then yields $\ddot{x} = -\vec{\nabla} \Phi$, as we expect from Newtonian physics.
The issue is that your metric doesn't do a terribly good job at explaining physics beyond Newtonian gravity, such as light-bending, time delay or perihelion precession.  A common way of looking at the weak-field metric for a gravity model is the Parametrized Post-Newtonian (PPN) formalism, which a large number of gravitational models reduce to in the weak-field case.  In this formalism, your metric has all PPN parameters equal to zero.  However, experiments show that the parameters $\gamma$ and $\beta$ are $1$ to within a few parts in $10^{-5}$.
