I would go for some analysis first.

Look at the symmetry, with an even density, it would resolve to the
field from the two circle in the vertical plane with the target point
and the center of the torus.

Looks like a 2 planets field, in 2D.

Compute some particular point to adjust the values.
Such points could:
 * center of torus (no gravity, hence both centers of mass are identical)
 * on the external circle on the middle plane (due to symmetry, it
should behave like the whole mass was at the torus center, allowing to
adjust the apparent mass of each "circle" to keep the previous equality)

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