NavList:

After going down a rabbit hole on this, I found this paper and a YouTube video on the subject by its author:

https://www.maa.org/sites/default/files/pdf/awards/college.math.j.47.2.95.pdf

https://www.youtube.com/watch?v=-Zaf99s5h84

With regards to geometric constructions, points on a logarithmic curve can be found through finding successive geometric means, but constructing the catenary itself requires first drawing a line segment of length e, possibly from working backwards from an actual hanging chain.

From the above paper, it seems Leibniz himself suggested using a physical hanging chain to determine ("natural," base-e) logarithms if one loses one's logarithm tables "during long journeys," but it doesn't seem it would be of much use in celestial navigation specifically.

Firstly, the technique requires hanging a chain between two horizontal points, making it impractical aboard a moving ship.

But more importantly, for celestial navigation logarithms are a computing tool for determining products and quotients of numbers (e.g. sines and cosines of angles), and sight reduction doesn't inherently require logarithms.  If you lose your tables of logarithms but not your tables of (natural) trigonometric functions, it seems more practical to resort to a multiplication table.  And if you lose all your tables, latitude and longitude can be determined through geometric constructions directly; precision would be limited by the scale of your drawing, but the same is true of geometrically constructed logarithms.

For those interested, I found a presentation and associated slides on the actual geometric construction of a catenary curve:

https://www.youtube.com/watch?v=TZ6mjPbmUyU

http://mikeraugh.org/Talks/JMM17B.pdf

And if you're like me and don't know how to construct a geometric mean (audio is unnecessary):

https://www.youtube.com/watch?v=PuHw0tUPPPY