Lunar atmospheric tides are likely insignificant for weather, although Guoqing (2005) asserts that The lunar revolution around the earth strongly influences the atmospheric circulation. They don't seem to be studied terribly much. What insights can we gain from the study of lunar atmospheric tides?
This is a partial answer detailing some of the recent research in lunar atmospheric tides. Many of which go beyond the 27.3 and 13.6 day cycles discussed in the Guoqing, 2005 paper, referenced in your question. There has been some considerable research in this in recent years (an example is this list from Google Scholar of research since 2013).
A couple of examples are explained below that show that all manner of lunar cycles seem to have an effect on the Earth's atmosphere, generating a complexity of tides that tidally affect all layers of the atmosphere. As we've been able to take long term measurements of various atmospheric parameters (most notably temperature, pressure and velocity) we have been able to come across patterns that suggest that several lunar cycles have made an imprint on them - in a way, suggesting that all lunar cycles have an affect on our atmosphere.
The lunar semi-diurnal cycle has been found to have a definite signature in the amplitude and phases of the temperature field of parts of the atmosphere between 20-120km altitude, according to research presented in the article A global view of the atmospheric lunar semidiurnal tide (Paulino et al. 2013) with the most pronounced signature observed in the lower thermosphere region.
Observations made between 1947-1994 in the article Long-Term Lunar Atmospheric Tides in the Southern Hemisphere (Wilson and Siderankov, 2013) have detected 4 roughly equidistant standing-wave summertime patterns with mean sea level pressure (MSLP) and sea-surface temperature (SST) consistent with being influenced by the 18.6 year Draconic lunar atmospheric cycle. Superimposed on these is evidence of the SST's being affected longterm by the Perigee-Syzygy (or 'supermoon') lunar cycle.
A consequence is that the standing-wave pattern of the MSLP will interact and cause abnormal atmospheric pressure when passing over the semi-permanent South Pacific subtropical high roughly every 4.5 years, the consequences are that these affect the strength of the East-Pacific trade winds and according to the authors, may lead to conditions that "preferentially favor the onset of La Niña/El Niño events."