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Paul,

I've been told that taking sights in moon-shine is risky, because you cannot be sure that it is the real visible horizon that you see. Whether that is true or not I cannot judge.

The rank of sub-lieutenant may be a translation error, from French via Swedish to English. The Swedish term in my book is "löjtnant". 

Anyway, I got one step, or giant leap maybe, forward by finding out what those arcs are used for. In "Bulletin of the American Mathematical Society", volume 31, number 1-2, January-February 1925, there is a review of H.B.Goodwin's "The Alpha, Beta, Gamma Navigation Tables", London 1921. The reviewer describes how the tables are designed, and following this description I have changed your proposed formula to be based on sines instead of haversines and can recreate Gerbault's arc calculations. The reviewer talks about cos formulas but sine and cosine are not that different, it is more a question of reading a table from down up instead of from up down. My findings attached. Still the logs have to be explained.

Lars