It depends what you mean by 'Would remain the same'. If you are asking about the exact precise orbit, the answer is yes, it would change, since the Schwarzschild (and also Kerr solution of a rotating black hole) solution is a vacuum solution of the Einstein equations, so it describes strictly speaking only Vacuum, with a curvature singularity in the centre. Since the Sun is made out of matter, has a finite size and you need many, many variables to fully describe it (for example it is non-spherical) the metric around it is not Schwarzschild (remember that a Schwarzschild black hole is only described by one parameter).
However, the Schwarzschild metric is a good approximation around the Sun, and for current observational purposes it describes nature well. What one should do, is to calculate the perturbations around the Schwarzschild metric in some form of power series in the characteristic length scale of the Sun over the Schwarzschild radius of the Sun. You will then find that these corrections are negligible at our current measurement capabilities. This last statement is with the exception of the advancement of the perihelion of Mercury, where we can now measure the quadrupole moment $J_2$ of the sun to be non-zero. This means that one should replace the Sun not with a Schwarzschild black hole, but with a Kerr black hole (to achieve a non-vanishing quadrupole moment).
As an aside, it is also important to note that the parameter that describes a Schwarzschild black hole $M$ is precisely constructed such that the physics around that central object reduces to the orbits of a body with mass $M$ in Newtonian gravity. It is therefore not surprising that the orbits do not change that much far from the event horizon.