The Einstein field equations (EFE) need to be modified to support the new metric. There's no way around that, because as you say, the Schwarzschild solution derives unambiguously from the EFE, for Schwarzschild geometry. I don't have a proposed modification to the EFE; that's beyond my skill level. However, I do know they must be modified if the theory is to be scientific, because (and this is the motivation) I can prove that the Schwarzschild metric is inconsistent with GR as a whole. I can elaborate on that if you want.

In my opinion, your modified solution does not remove the singularity from the Schwarzschild solution ...

To be clear, you're talking about your change to the new metric, right? I said before that the new equation for escape velocity, derivable from the new metric, is:

v=\sqrt{\frac{r_s}{r+r_s}}

In that case there needn't be singularities. A body contained at r = 0 (a singularity) can be relegated to a limiting case only, when the escape speed is otherwise < c. With the new metric, unlike Schwarzschild's, there is no case where a body need implode to r = 0.

I leave SR intact, and fully interpret it. I say that GR incorrectly incorporates SR, leading to the inconsistency mentioned above.

However, if your new metric can in principle account for the rotational curve evidence for DM it may have some interest -- I don't think this is the only evidence it would have to account for.

Note that the link I gave you shows 2 kinds of observational evidence for DM. The new metric can explain both of them. The new metric also makes the following major problems of physics vanish: GR's incompatibility with QM in regards to singularities, and the black hole information loss paradox.