When I was an undergraduate, some friends were involved in a proposal ("project SEE") to do exactly this, albeit with smaller masses. The proposed method was a satellite specially crafted to produce negligible gravitational forces on the objects inside (by having the satellite's mass distribution match the charge distribution on the surface of a conductor of the same shape, since electric and gravitational fields both go like $1/r^2$). Then test masses would be released in a cavity inside the satellite, where the dominant gravitational interactions would be with the Earth and with each other. There is a neat effect where the gravitational interaction between co-orbiting satellites is effectively repulsive, leading to horseshoe orbits.
Needless to say, the number of "secondary effects" was very large.
As a fundamental test of gravitation, the project never launched.
A successful project which would interest you was the Gravity Recovery And Climate Experiment, GRACE. This was a pair of identical satellites in nearly-polar pursuit orbits, separated by about 200 km; the separation between the satellites was continuously measured to very high accuracy. The dominant effect on those orbits was density fluctuations in the material near Earth's surface. As the two satellites approached a dense region of Earth, they'd be attracted to it and speed up, with the leading satellite pulling away; as they receded from the higher-density region, the gap would close back again. The GRACE orbit covered most of Earth's surface frequently (daily?), so in addition to being sensitive to geological structures, they were able to measure things like the seasonal movement of groundwater and glacial ice.
I think that's one of the problems that doomed Project SEE. I remember sitting in a lunch meeting where that group revealed their design sensitivity was precise enough to locate a herd of buffalo, and wondering how they would distinguish a novel gravitational effect from a business decision made by a rancher in rural nowhere. But there were potential issues with the collection of the position data, with the thermal management, etc. as well.
This is more of a pointer to the literature than an answer to your question. I do think there's an obvious flaw in your proposal to have two one-ton objects orbiting each other with a center-of-mass separation of ten centimeters, which is that I don't know of any material that's dense enough for that to physically take place. Such a direct two-body orbit between such small masses would also be painfully slow, so that you probably couldn't ignore terrestrial/solar/jovian gravity unless you conducted the experiment way out in the outer solar system.