We are straying far from my area of expertise. :-) (except at the end).

> Mercury, again, might be a good case

In our solar system I'd look to Jupiter, whose magnetic strength is tens of thousands of times stronger than Earth's, and which has lots of bodies of various sizes orbiting it.

There are magnetic interactions with several of the moons explored by recent space probes, but they are very small. Ganymede is probably the most interesting case. I would be surprised if there were significant orbital perturbations from magnetics on the Galilean moons, or even the mid-size ones.

Smaller, especially grain-sized bodies, I don't know. I know next to nothing about Jupiter's rings, but on general chemistry grounds there will be electromagnetic interactions with the planet's magnetosphere. (I suspect these are dwarfed by radiation pressures though; the biggest forcing radiation will be tangential to the ring-components' orbits around Jupiter, and as those objects are small the effect will be relatively large, causing illuminated grains to tend to spiral inwards to Jupiter, but how long does a ring particle stay illuminated rather than in shadow? And there are likely forces that push grains outward, too). None of this likely needs corrections to classical Newtonian gravitation. (The authors of the paper linked at the top do not consider any of the solar system's moons or smaller objects).

I'd start here <https://en.wikipedia.org/wiki/Magnetosphere_of_Jupiter#Inter...>, notably the paragraph starting, "The icy Galilean moons, Europa, Ganymede and Callisto, all generate induced magnetic moments in response to changes in Jupiter's magnetic field", and visit the various citations in that paragraph and the following two.

> evolution of small parameters

You've hit on the right word, "evolution", which is much more important than "small". Can we apply perturbation theory? Do we get a nice Taylor-series-like expansion where higher order terms can be treated as marginal or negligible, in the style of Wilsonian effective field theory? See Clifford Will's article, <https://www.pnas.org/doi/pdf/10.1073/pnas.1103127108>, which is probably close to accessible. It is mainly interested in "quite relativistic" systems, however, and we do not have those in our solar system. Almost nothing in the solar system needs even 1PN (see what that means in the article, or in e.g. the wikipedia article on the post-Newtonian expansion) correction. The work linked at the top is neat because PN effects may actually matter for the ultimate fate of Mercury.

Finally, this beautiful paper by Soffel (Past President of IAU Commission 52 Relativity in Fundamental Astronomy) discusses the (ir)relevance of higher-order post-Newtonian corrections to the Earth's gravitational field. <https://link.springer.com/article/10.1007/s00190-016-0927-4> (find it on sci-hub.ru). ("Above, we have demonstrated that the first post-Newtonian approximation is completely sufficient for a description of the Earth's gravitational field and also for a description of satellite orbits."). Table 3 lists the various accelerations on the LAGEOS satellite. Solar radiation pressure, thermal re-radiation, and even albedo pressure (at 3.5e-19 m/s^2) are in the table; nothing like like "magnetotorquer" effects even appears.