I checked NYC Parks dept and found the following about Great Kills harbor:
"# All parts of the mooring, including the buoys, anchors and chains, shall be supplied and installed by the permittee and shall remain the property of the permittee at the mooring fields at Sheepshead Bay and Great Kills Harbor.
# Moorings shall be inspected for deterioration at least every two years and repaired or replaced if necessary. The Department may require, as a condition of renewing a permit, evidence that an inspection has been made, including a description by the person who made the inspection of the condition of the mooring and the qualifications of such person to make such inspection."
I think you need to ask around at Great Kills to find out who has the equipment for installing/fixing/removing moorings there. It will take a significant work boat or barge to do this kind of work properly.
I'm not sure how the water visibility is in Great Kills harbor but it may be possible to have a diver inspect the mooring & chain. Again, for this information you will have to ask around with other boat/mooring owners in Great Kills harbor; preferably someone who has kept a boat there for at least 5 years or more.