You have taught yourself through your own experimenting and curiousity a famous lesson. You are indeed doing exactly as Laplace did and your findings are the same as Laplace's.

To add to [David Hammen's Answer](http://physics.stackexchange.com/a/186196/26076): David is correct that Laplace's attempts at introducing latency lead to an unstable solar system. But there is a way to mostly succeed with Laplace's method and that is to make the latency Lorentz invariant. That is, you begin with the electrostatic law (or its equivalent Gauss law for gravity), postulate that the Gauss form this holds for dynamic systems and find the simplest Lorentz-invariant tensor that contains your gravity field. The result if you begin with electrostatics is Maxwell's equations and the Lorentz force law; the mathematically wholly analogous result if you begin with Newton's gravitation is [Gravitoelectromagnetism](http://en.wikipedia.org/wiki/Gravitoelectromagnetism). Gravitoelectromagnetism is consistent with all the results of [Gravity Probe B](http://en.wikipedia.org/wiki/Gravity_Probe_B) and other results observed in the solar system (more on this [here](http://physics.stackexchange.com/a/121324/26076), [here](http://physics.stackexchange.com/a/178173/26076) and [here](http://physics.stackexchange.com/a/179252/26076)). In particular, although the orbits are still theoretically unstable, they are stable over timeframes many times the age of the universe for all observations made in the solar system. Gravitoelectromagnetism is thus an extremely accurate approximation of General Relativity for our solar system.

The main theoretical difference between GEM and GTR is the source terms. In Maxwell's Equations, the four-current density is the source and is a rank 1 tensor. In GEM, the analogous four gravitational-mass-current density cannot both be a rank 1 tensor *and* be consistent with the source term in GTR: the stress-energy tensor, which is rank 2. Experimentally, even though the instability in GEM for solar system situations is small enough to be consistent with observations, the instability is still much bigger than that foretold by GTR. We have an observation that can tell the two apart: the [Hulse-Taylor binary star system](http://en.wikipedia.org/wiki/PSR_B1913%2B16) whose spin-down owing to gravitational wave radiation is consistent with GTR but inconsistent with GEM.