Dave Walden you said:Using a two second theodolite from the beach.

That theodolite might give you a relative bearing to 2” accuracy, but how would you set it up?  If you used north, how would you find where north was to that degree of accuracy, a PIM or geo-theodolite possibly?  They’re a bit specialised, and I believe accuracy is very much related to price, which can be as high as $250,000 for a 2” one.  Alternatively, if you had some accurately surveyed points or a base line you could use relative bearings, but then you would know where you were already, so you wouldn’t need to use celestial.  North of course, can be found very accurately using a theodolite for celestial, but to calculate azimuth to 2” to measure north to 2”, you’d need to know where you were in the first place.  

Alternatively, could you make use of the relative bearing accuracy to take bearings on three stars with respect to each other and find the point where those three lines crossed as you might by taking bearings on three points with a hand bearing compass?  You don’t need an accurate north for that, you just need accurate angles in between.
 
Perhaps you could set up some equations such that you slewed the bearings equally until the cocked hat they produced was reduced to a minimum, a bit like correcting the clock in a GNSS receiver.

Another possibility might be to plot lines of constant azimuth difference.  Is it possible to rig up a programme to do that? DaveP