I am not aware of any public moorings at Lady Elliot. Having made perhaps a dozen trips to the Bunker Group, I have not encountered any moorings other than those for
commercial operations at Lady Musgrave. I believe your only option is to
anchor on the leeward side of the fringing reef which I would not recommend for an over night stay. Generally this would be the NW side. In a typical SE condition, this will be a little lumpy. You would be best to anchor overnight at Lady Musgrave
lagoon some 21NM to the NNW, which is an excellent anchorage, inside the large
lagoon. Good sandy bottom at about 6-8m with excellent visibility wrt bommies etc. You can then take a day trip to Lady Elliot and return. I've been in there with 40kts over several days and its generally very comfortable. The only other bullet proof anchorage is at Fitzroy some 20NM further north. Again, been there in fierce storms with good protection. Obviously there are lots of places to anchor on the leeside of
reefs, but you have to be prepared to ride out a
wind change as there is no safe relocation after dark in a blow in those waters.