That is what is done in KSP ever since planets beyond the Mun were added - on every frame, the coordinate system is re-centered on the active vehicle. The problem that still remains is mainly in trajectory calculations - you might have an intercept with a planet way on the other side of the solar system, and you can generally see that the predicted trajectory does not pass through the encounter point in these cases due to floating point inaccuracies.
Other stuff too if you want to look around, I did a quick and dirty solar system a couple of decades ago, one with texture maps where possible, and quickly discovered that Pluto was all lumpy .... simply because tessellating a sphere that far out quickly runs into FP resolution issues if the Sun is the origin