That idea is interesting. Thinking about gravitational redshift in terms of H, it would seem its value might be greater near a body, perhaps varying with the square of the distance, I haven't worked it out yet, and then decreasing in such a way to be proportional to the distance at long range. This might also be similar to that of the rotational speed of stars around a galaxy, although that also has a lot to do with the geometry of the galaxy itself. And perhaps also with something like the Pioneer anomaly. All of these seem to be related somewhat in such a way, having a large potential near a body and then eventually leveling off directly proportional to distance.

The thing is, though, when trying to work out what you were describing about two particles travelling away from a body at the same speed with some initial separation, that if we were to emit these particles in the same way as we might photons of light, from a common source at some distance from the center of the body, then the particles would only have the same speed at the source, but not at some separation between them. So the first particle would be emitted directly away from the body at v, and say a second later, the other at the same original speed. During the difference in the time of emission, the first will have slowed somewhat. This will lessen the separation between them, not lengthen. As the particles continue to travel, the second will always have a slightly greater speed than the first, even though the gravity upon it is greater, since the first has already experienced that same gravity earlier, being a step ahead, and has slowed down more accordingly. Therefore, the separation between the two will continually shorten, not lengthen.

It is just the opposite of dropping two particles toward the body from a distance, but with some time between each drop. The first to be dropped will have undergone some acceleration due to the gravity before the second is dropped, and so will always be travelling faster than the second, the separation between them increasing steadily all the way down, and similarly then, decreasing on the way up.

Perhaps some slight variation on that idea might work out, but it looks like one would be walking a fine line in order to do so.

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