If I understand you correctly, what you're trying to do won't work.

In the center of gravity frame, gravity will act on the center of mass as if all mass was concentrated at that point.

In short, you need a 3rd body - then can use Kepler's laws to find the center of mass motion around the 3rd body.

For instance, the center of mass of the Earth and the Moon follows an elliptical orbit around the Sun.

Both the Earth and the Moon wobble around the center of mass as it follows the elliptical orbit.

If I understand you correctly, what you're trying to do won't work.

In the center of gravity frame, gravity will act on the center of mass as if all mass was concentrated at that point.

In short, you need a 3rd body. Then you can use Kepler's laws to find the center of mass motion around the 3rd body.

For instance, the center of mass of the Earth and the Moon follows an elliptical orbit around the Sun which can be calculated using Kepler's laws.