From my layman's perspective it seems that a cheap, quick, and archaeologically sound approach would be to line out the existing hole with geotextile (permeable membrane) and then fill the hole with chalk. That way the geotextile would form a boundary layer to differentiate the fill material from the original, but the general structure of the hill would have been restored. In the future, if someone comes up with a better way to do it, it's a fairly easy job to dig out the chalk fill and remove the membrane.