Please note none of the following is legal advice as such. If you want that talk to a Solicitor/Lawyer.
Questions
Is it legal to pan and sluice in the river?
I would say NO based on parts of the National Parks & Wildlife Act + Regulation
i.e.
There is a section of the Act specifically addressing the sign off on plans of management that include waterways - 80 Lands submerged by water
In the Regulation it says you must not "(h) carry or possess,interfere with, dig up, cut up, collect or remove for any purpose any soil, sand, gravel, fossil, clay, rock, ochre, mineral, timber (whether or not consisting of or including dead timber), gum resin, humus or other natural substance or object in a park, whether on land or on or under water"
The park you have put up here is a designated wilderness area:- ""wilderness area"means land (including subterranean land) that is reserved under this Act and is declared to be awilderness areaunder theWilderness Act 1987"
NSW NPWS also manage a lot of "special" areas in partnership with NSW Water that are either completely off limits or limited/restricted access.
There are also other areas such as Karst that includes subterranean land, wild rivers etc.
IMO it looks pretty clear that NPWS have covered land under water + waterway management within their parks.
Can a prospector transit with sluicing equipment through the National Park to get to and then leave the river?
I believe any contraband or illegal activity type equipment can only be carried through National Parks in NSW & not if you plan to stop in the park. This doesn't appear to be specific to fossicking/prospecting equipment although Rangers do have powers to search & seize in regards to alleged illegal activities.
If otherwise illegal, if the prospector navigates the river (from outside the National Park boundaries) and does not set foot either side of the river (e.g. only standing on bed rock in the middle of the river) is it then legal?
NO