NavList:

What you have described is the core idea of moon culminations, an old
method to determine longitudes of fixed points. The largest error came
from the Moon ephemeris, so transits were also observed from a point of
known longitude. That cancelled the ephemeris error, and you ended up
with the longitude difference between the observatories.

Lunar culminations generally had lower accuracy than the mechanical
solution of transporting sets of chronometers back and forth between the
observatories.

This 1870 report about determining the longitude of Wellington covers all of your points. Lunar culminations, using the moon ephemeris as an interim solution, observations at a point of known longitude (Greenwhich), transporting chronometers, and the electric telegraph (within NZ). Lunar culminations were considered inferior to transporting chronometers. There is no mention of Lunars.

https://atojs.natlib.govt.nz/cgi-bin/imageserver/imageserver.pl?oid=AJHR1870-I.2.2.4.32&getpdf=true