I saw another explanation as to why women shouldn’t say kiddush levanah (which I reject):
The Magen Avraham paskens according to the view of the Shela I gave above; the fact that women don’t make the brakha because of Hava’s sin amounts to the klal of Ein Kateigar Na’aseh Saneigar.”
The Debretziner Rav (Be’er Moshe 6:135) says that women light candles on Erev Shabbat to be metaken Hava’s sin. Why would women be metaken one sin they caused, and not be metaken another sin (by reciting kiddush levana)?
After offering an excursus on Ein Kateigar Na’aseh Saneigar, he says that since kiddush levana involves making a Bracha on the actual moon that Hava caused harm to, it is not proper for women to make a bracha. However the candle only symbolizes the Neshama and is not the Neshama itself therefore the Klal of Ein Kateigar Na’aseh Saneigar does not apply and by lighting the candles it rectifies their mistake.
I remain steadfast in my position that women ought to make some form of kiddush levana, as every opposing answer I see is mystical and not legal in nature.