I think you overestimate our technical abilities, if you think we should, with current tech, be able to measure such differences.
No?
I was thinking of actually measuring the motion of Neptune against the background of distant galaxies. The Hubble telescope or alternatively, the James Webb or Large Synoptic Survey telescopes (once working) could take photographs (wavelength dependent on telescope) of it at regular intervals over 6 or 8 months of each year. The timing of the photographs would need to be precise, but they could be done with an atomic clock or alternatively, many photographs could be taken over an extended time, but an atomic clock might need to be used to determine exactly when they were each taken. Since the data might need to be collected over many decades, it might take a long time to get a precise answer. But, even early observations would give us a good idea, and it should be technically feasible. There would be problems with the brightness of Neptune versus the brightness of background galaxies, retrograde motion, wobble due to Triton, calculating how much DM affects the Earth itself, etc., but nothing that is technically or mathematically insurmountable.
As for your calculation of Earth, the average diameter of Neptune's orbit around the Sun is ~9 billion km. It is nearly a circular orbit as well (hence, the acceleration / deceleration due to the sun's gravity is very tiny). The amount of dark matter in a spherical halo within the galactic radius when it is closest to the galactic core versus when it is away from the galactic core would be ~99.999990 percent (assuming an equal distribution, a more densely packed distribution would result in a higher percentage). The distance to the galactic core from closest to furthest would be ~99.9999963 percent. But by collecting a lot of data over large time scales, the precision of the measurements increases (by comparing data collected a long time ago with recent data).
Now, this assumes that the sun doesn't significantly change its distance from the galactic core over these time frames. Considering that it takes the sun 225+ million years to orbit the galaxy, even if it has an elliptical orbit, the change in distance would be extremely small in a time frame of decades. But, it should also be measured as best as possible and taken into account for the calculations.
This also wouldn't prevent us from measuring Mars (or other planets) instead, or in addition. The amount of time observing for Mars would be a lot less (3 to 4 years, maybe double that to get more data), but the precision would need to be a lot greater (probably at least two orders of magnitude). Course, since Mars is closer, precision might be greater because of it.
If we can measure that supernovas are accelerating away from our galaxy based on extremely precise temporary brightness from them and hence, calculate that 72% of the universe is Dark Energy, we can measure this. It's a lot closer, easier to see, we can view at optimal times, etc.
Note: I did not state that such an experiment would not be expensive or time consuming, I merely stated that it should be technically feasible. There are also faster and more precise, but even more expensive but technically feasible ways, to measure this (e.g. multiple telescopes in orbit on various sides of the Earth viewing the planet with different angles to acquire a more precise change in background, distance from Earth/sun, etc.). But, it should still be doable with technology already built or in the works today.