This is not a complete answer. It is instead an extended comment to the following:

I understand I will have to retard the gravity from each source by its particular light-time, as well as correct for the light time for the HST images.

While you do want to correct for light time travel with regard to seeing a moving remote object, you definitely do not want to do the first part (retarded gravity). That is not how Newtonian mechanics works, nor is it not how general relativity works.

There is no lag in Newtonian mechanics; gravitation is instantaneous in Newtonian mechanics. In general relativity, there are some terms that act like lags, but there are other terms that act like leads. Those lag-like and lead-like terms in general relativity nearly cancel for small gravitational sources such as our Sun. That near cancelation is what makes Newtonian mechanics very close to correct. Keep in mind that the even for Mercury, the relativistic effect is very, very small, 43 arc seconds per century or precession that is not explainable by Newtonian mechanics.

A couple centuries ago, Laplace investigated whether gravitation is instantaneous. He found that adding any significant lag to Newtonian gravity makes the solar system become unstable in short order and this concluded that the speed of gravity had to be very high, at least $7\times10^{6}$ times the speed of light. A couple of decades ago, another highly respected astronomer published a paper in Physics Letters A (a highly respectable physics journal) that came to the conclusion that the speed of gravity is at least 20 billion times the speed of light.

Both Laplace and this more recent author were wrong. Laplace can be forgiven for not having possession of a time machine that would carry him a century into his future. The latter author cannot. His article has been cited 175 times (per google scholar), but almost all of the citations are essentially "You're wrong. So very, very wrong, and here's why ..."

  The "here's why" is that that is not the way general relativity works.

The easy way to solve what you are trying to do is to ignore relativistic effects. Just assume Newtonian physics, in which gravitation is instantaneous (but the speed of light is not).

The hard way is to incorporate general relativity, to some extent. You'll need a relativistically-correct time scale (e.g., JPL's T_eph) and some kind of post-Newtonian model of gravitation that is, at least to first order, consistent with general relativity. Do that and you'll be on par with the groups that develop extremely accurate solar system ephemerides.

This is not a complete answer. It is instead an extended comment to the following:

I understand I will have to retard the gravity from each source by its particular light-time, as well as correct for the light time for the HST images.

While you do want to correct for light time travel with regard to seeing a moving remote object, you definitely do not want to do the first part (retarded gravity). That is not how Newtonian mechanics works, nor is it not how general relativity works.

There is no lag in Newtonian mechanics; gravitation is instantaneous in Newtonian mechanics. In general relativity, there are some terms that act like lags, but there are other terms that act like leads. Those lag-like and lead-like terms in general relativity nearly cancel for small gravitational sources such as our Sun. That near cancelation is what makes Newtonian mechanics very close to correct. Keep in mind that the even for Mercury, the relativistic effect is very, very small: Only 43 arc seconds per century of precession that is not explainable by Newtonian mechanics.

A couple centuries ago, Laplace investigated whether gravitation is instantaneous. He found that adding any significant lag to Newtonian gravity makes the solar system become unstable in short order and this concluded that the speed of gravity had to be very high, at least $7\times10^{6}$ times the speed of light. A couple of decades ago, another highly respected astronomer published a paper in Physics Letters A (a highly respectable physics journal) that came to the conclusion that the speed of gravity is at least 20 billion times the speed of light.

Both Laplace and this more recent author were wrong. Laplace can be forgiven for not having possession of a time machine that would carry him a century into his future. The latter author cannot. His article has been cited 175 times (per google scholar), but almost all of the citations are essentially "You're wrong. So very, very wrong, and here's why ..." The "here's why" is that that is not the way general relativity works.

The easy way to solve what you are trying to do is to ignore relativistic effects. Just assume Newtonian physics, in which gravitation is instantaneous, but the speed of light is not.

The hard way is to incorporate general relativity, to some extent. You'll need a relativistically-correct time scale (e.g., JPL's T_eph) and some kind of post-Newtonian model of gravitation that is, at least to first order, consistent with general relativity. For example, see The Planetary and Lunar Ephemerides DE430 and DE431. Do that and you'll be on par with the groups that develop extremely accurate solar system ephemerides.

This is not a complete answer. It is instead an extended comment to the following:

I understand I will have to retard the gravity from each source by its particular light-time, as well as correct for the light time for the HST images.

While you do want to correct for light time travel with regard to seeing a moving remote object, you definitely do not want to do the first part (retarded gravity). That is not Newtonian mechanics works, where gravitation is instantaneous, and it is not how general relativity works, either.

A couple centuries ago, Laplace investigated whether gravitation is instantaneous. He found that adding any significant lag to Newtonian gravity makes the solar system become unstable in short order and this concluded that the speed of gravity had to be very high, at least $7\times10^{6}$ times the speed of light. A couple of decades ago, another highly respected astronomer published a paper in Physics Letters A (a highly respectable physics journal) that came to the conclusion that the speed of gravity is at least 20 billion times the speed of light.

Both Laplace and this more recent author were wrong. Laplace can be forgiven for not having possession of a time machine that would carry him a century into his future. The latter author cannot. His article has been cited 175 times (per google scholar), but almost all of the citations are essentially "You're wrong. So very, very wrong, and here's why ..."

The "here's why" is that that is not the way general relativity works

The easy way to solve what you are trying to do is to ignore relativistic effects. Just assume Newtonian physics, in which gravitation is instantaneous (but the speed of light is not).

The hard way is to incorporate general relativity, to some extent. You'll need a relativistically-correct time scale (e.g., JPL's T_eph) and some kind of post-Newtonian model of gravitation that is, at least to first order, consistent with general relativity. Do that and you'll be on par with the groups that develop extremely accurate solar system ephemerides.

This is not a complete answer. It is instead an extended comment to the following:

I understand I will have to retard the gravity from each source by its particular light-time, as well as correct for the light time for the HST images.

While you do want to correct for light time travel with regard to seeing a moving remote object, you definitely do not want to do the first part (retarded gravity). That is not how Newtonian mechanics works, nor is it not how general relativity works.

There is no lag in Newtonian mechanics; gravitation is instantaneous in Newtonian mechanics. In general relativity, there are some terms that act like lags, but there are other terms that act like leads. Those lag-like and lead-like terms in general relativity nearly cancel for small gravitational sources such as our Sun. That near cancelation is what makes Newtonian mechanics very close to correct. Keep in mind that the even for Mercury, the relativistic effect is very, very small, 43 arc seconds per century or precession that is not explainable by Newtonian mechanics.

A couple centuries ago, Laplace investigated whether gravitation is instantaneous. He found that adding any significant lag to Newtonian gravity makes the solar system become unstable in short order and this concluded that the speed of gravity had to be very high, at least $7\times10^{6}$ times the speed of light. A couple of decades ago, another highly respected astronomer published a paper in Physics Letters A (a highly respectable physics journal) that came to the conclusion that the speed of gravity is at least 20 billion times the speed of light.

Both Laplace and this more recent author were wrong. Laplace can be forgiven for not having possession of a time machine that would carry him a century into his future. The latter author cannot. His article has been cited 175 times (per google scholar), but almost all of the citations are essentially "You're wrong. So very, very wrong, and here's why ..."

The "here's why" is that that is not the way general relativity works.

The easy way to solve what you are trying to do is to ignore relativistic effects. Just assume Newtonian physics, in which gravitation is instantaneous (but the speed of light is not).

The hard way is to incorporate general relativity, to some extent. You'll need a relativistically-correct time scale (e.g., JPL's T_eph) and some kind of post-Newtonian model of gravitation that is, at least to first order, consistent with general relativity. Do that and you'll be on par with the groups that develop extremely accurate solar system ephemerides.