I can't just pull this out of my hat, so I went to do some research, and it turns out that this is a highly non-trivial problem that was only solved in 1975 ( I think ); I won't even attempt to do this myself, not even for something comparatively simple as Schwarzschild. In any case, the answer seems to be no, the electric fields aren't the same. This makes sense if one just looks at the form of the Maxwell equations :

The Hodge dual depends explicitly on the metric, and the differentials in the exterior derivative become covariant derivatives, so the resulting system of differential equations will depend explicitly on the Schwarzschild r coordinate. Even though the charges as described by J are identical, the resulting electrostatic fields are position-depended, since they are distorted by the curvature of space-time, which is not constant.