The other key reference on this is by Bush and Jackson "Correction of spherical error of a pendulum" in
Volume 252, Issue 6, December 1951, Pages 463-467
alas only available if you pay or have institutional access. Vannevar Bush was an eminent US science maven in WW2 and after who played a major role in things like early computers etc. Also interested in clocks and there are a couple of articles in Scientific American describing their experiments dating from the early 1960s. The article above has a reference to Loseby but he only had the rather negative Grimthorpe reference and hadn't seen the Practical Mechanics stuff and the test report by Airy. The scheme used a tension spring pulling downward on a point on the pendulum, the other end being held vertically below the point of suspension. As the pendulum swings the spring is slightly stretched and the geometry can be designed to provide an additional restoring force proportional to the cube of the angle that can cancel the cubic term of the sinusoidal restoring force. In practice they used a cantilever leaf spring at right angles to the pendulum with a thin wire connecting it to the pendulum rod. Good results are reported, I haven't compared them with Airy's measurement yet.
As far as I can see Loseby's spring creates the cubic dependency through its shape and from my brief look at the description he doesn't say what this has to be mathematically. By contrast Bush only assumes that the spring is linear (a good approximation for small extensions) and the cubic term comes from the geometry.