This is a good question. You are reading the end of the statement from the Mishnah Berurah backward.
דהם גרמו פגם הלבנה doesn't mean that the women caused, "garmu", the "blemish" in the moon, meaning its reduction. The plural personal pronoun for women is not “Hem" הם. It is “Hen" הן. So the statement cannot, according to the grammar, be referring to women.
What "hem" is referring to is the "actions", meaning מעשה, from מצות עשה, mitzvot of action or positive mitzvot. The plural would be מעשיות.The word "ma'aseh" is a masculine noun and in plural as a personal pronoun is "hem".
There are 248 positive commandments which correspond to the 248 limbs of the male body (Mishna Ohelot 1:8). Women on the other hand have 252 limbs according to Sefer Raziel or 251 according to the Mishnah Torah: Hilchot Tumat Meit 2:2. The 248 mitzvoth of action specifically relate to the male body and the masculine. Males are obligated in them all.
There were no mitzvot being performed at the time of the formation of the sun and the moon, meaning the two great luminaries. This happened on the fourth day (Bereshit 1:14) before the creation of HaAdam, which was on the sixth day (Bereshit 1:26). Therefore, "Garam" must be understood differently.
"Garam" is not, in this sentence from the Mishnah Berurah, a verb as in "garmu", but rather is a modified noun. The word "garam" is modified to show it is associated with a masculine noun. The noun, "HaLavanah" is feminine. It is thus associated with the noun "pagam" which is masculine. They, meaning women, are the effect of the "blemish" of the moon, meaning it's reduction and youth in comparison to the larger and older luminary, meaning the sun.
It means that the "blemish" of the moon, meaning its reduction, is the cause, pronounced “garmo", of women and also these mitzvot of action being discussed.
And that is also the meaning of the phrase, "מצות עשה שהזמן גרמא". That the creation of time, meaning linear time, what we perceive as past, present and future, also called (זמן משוער) as discussed in Sefer HaIkkarim of Rabbi Yosef Albo, Second Discourse, Chapter 18, is the cause of these mitzvot of action.
Linear time came into being with the creation of the two great luminaries and the reduction of the moon. This is the emphasis of Bereshit 1:14 which says that the sun and moon are for seasons, new moons, days and years.
On the first day of creation, there was a revelation of a single light with no luminary source (Bereshit 1:3). This is an expression within creation of the concept that the Creator has no cause or source. (Mishnah Torah, Hilchot Yesodei HaTorah 1:1, 1:7, 1:11)
Time at that stage of creation was not measured. It was unmeasured time which is called (זמן בלתי משוער) as discussed in the Sefer HaIkkarim cited above. This means that past, present and future were at least potentially simultaneous. It is more like the idea of undifferentiated “existence” or in the words of Rambam, מצוי ראשון.
At a later stage in the creation, that light was divided into two separate components (Bereshit 1:14). This process related to what the Creator's original thought was in performing the creation. This process of starting as one and dividing into two separate parts was to bring into actualization the first thought of creation (Yafeh Toar to Bereshit Rabbah 8:1, Midrash HaGadol 2:22 in the name of Rabbi Abahu).
The "great luminaries", what would become the sun and the moon, were created on the 4th day. In their initial creation, they were of equal size, meaning they were indistinguishable. As recorded in Midrash and explained in Rashi's commentary (Bereshit 1:16), the moon made a complaint to the Creator that there could not be two Rulers and a single crown. If they remained indistinguishable, it would imply duality in regard to the Creator. (Mishnah Torah, Hilchot Yesodei HaTorah 1:6-7)
This complaint is based upon the contradiction of the concepts of “one”, (אחד), like "HaShem is one", and “other” (עוד). But the requirement is to know and proclaim “There is no other apart from Him", (אין עוד מלבדו),when manifest in creation.
For G-d,"the Infinite" meaning אין סוף, these are not a contradiction. That is because G-d is without limitation, בלי גבול. But for the creation, meaning finite beings, בעלי גבול, these two concepts are contradictory and impossible to bear.
As a consequence, the Creator reduced the size of the moon and made it the "smaller or younger light” (Bereshit 1:16). The "greater or older light" would have dominion over the day and the ""smaller light" would have dominion over the night. In context, it is worth pointing out that "greater" and "smaller" can also mean chronological maturity in Hebrew. And this would also agree with our understanding of when the sun was formed as compared to the moon. The moon is younger.
It is important to note that this change in the relationship between the sun and the moon, that they were no longer considered indistinguishable and identical, that the sun was now considered senior, meaning older than the younger moon, had a consequence.
When the sun "speaks", meaning when it shines, the moon is in silence. It is not seen. And because the moon is younger, it listens to, learns from and repeats the teachings of its elder, the sun. This is the idea that the light of the moon is actually a reflection of the suns light and is only seen when then sun is not present. This is the concept of Mashpia and Mekabel, Teacher and Student as well as Parent and Child.
This event set up a paradigm in the order of creation which later had impact on men and women.
On the sixth day, the initial creation of the human being being was androgynous (Bereshit Rabbah 8:1 in the name of the Tanna, Rabbi Yirmiyahu ben Elazar based upon Bereshit 1:27, See also Bereshit Rabbah 17:2 in the name of the Amora, Rabbi Chiya bar Gamda). This is why the name “HaAdam” (האדם) is associated with the phrase, "דומה לעליון", similar to "Above", meaning similar to the Creator (Bereshit 1:26). A single creature contained the potential for both the male and the female. They were like two limbs in one body. This is a manifestation of the same pattern which began with the single light of the first day. It was at this time that G-d gave HaAdam his commandments (Bereshit 1:15-17). And one of them was the prohibition to eat from the tree.
Following this, the Creator separated these two limbs into separate components like the formation of the two great luminaries from the one light of the first day and built the first woman from one of them (Bereshit 2:21-22). There was a division into two indistinguishable parts and then the portion that was to become the first woman was built, meaning modified. The first woman became like the moon, and was considered junior to the first man.
And the first man, Adam, who had not been modified and retained his knowledge of the commandments which the Creator had given him, taught the woman, Chava, what the Creator expected from them. And because he feared that she would not listen to him, he added to G-d's prohibition and told her that they could not eat or touch the tree of knowledge because they would die as a consequence (Ikkar Siftei Chochomim 4 on Bereshit 3:3). And this is exactly what she repeated to the serpent when it asked her what the Creator required of them. And the serpent created doubt in her mind by pushing her against the tree. She discovered that she did not die from contact with the tree. And that doubt opened her to the possibility of the serpent convincing her to eat.
Her transgression was caused through her requirement to listen to her elder, her teacher, the first man, Adam. And this requirement was a consequence of the reduction of the moon in regard to the sun.
Following the events of the sin of the Tree of Knowledge of Good and Evil, the first woman was assigned by the Creator that relative position which was started by the moon (Bereshit 3:16).
And this is why women generally do not practice the sanctification of the moon as cited by the Shnei Luchot HaBrit, Sha'ar HaOtiot, Ot HaKuf-section Kedusha, with the words beginning, "There is a mitzvah which is dependent upon the mouth, and the eyes and the feet..." That from this reason (that the words of the kiddish levanah remind us that the moon was diminished and that in the future its light will be returned to be like the light of the sun) women distance themselves from this positive commandment even though they fulfill many commandments that are caused by time, for example shofar and lulav. We have never seen women fulfill the sanctification of the moon even though they are scrupulous in regard to all the prayers. Because the diminishment of the moon was the cause of the first woman, meaning Chava's sin. And they distance themselves because of embarrassment..."
The words of the Shnei Luchot HaBrit are exactly as explained above.
The emphasis is that a large part of a woman's being, her constitution, is connected to the influence of the cycle of time and specifically to the lunar cycle. It is for this reason that, in general, women are not obligated in positive commandments which are time bound or "caused by time".