Would it be a better answer to suggest a competition for firms to remove the material, and giving the winning contractor ownership of the material, and to award on a best value basis? Perhaps the best value proposal will be for one dollar.
A one-dollar contract is not quite a no-cost contract, but I think it is what the questioner is looking for.
It's simple -- an acquisition to remove debris, with price as the sole or major factor, and a hope that the price for the actual work will be offset by the residual or recycling value of the removed material. If the material really has value, a one-dollar proposal is possible.
There seem to be two basic questions here. The first is whether removal of construction-related debris should be accomplished using a construction (FAR Part 36) contract or a services contract (Part 37) contract. A services contract would be appropriate for this effort. FAR Part 2 defines construction as "construction, alteration, or repair (including dredging, excavating, and painting) of buildings, structures, or other real property." Removal of debris is not "construction, alternation, or repair."
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The second question appears to be whether to pursue a no-cost contract to remove "as much material as possible," or a best value contract to remove all of the material. This is a call for the customer (Civil Engineering) to make. The advantage of a no-cost contract is, obviously, no cost to the Government. The disadvantage is that not all of the debris may be removed, based on the offerors' evaluation of the cost-benefit of the project. The advantage of a best value award would be removal of all debris. The disadvantage is that it would entail a cost to the Government.