I started to write this last night, but when I got up to research the lunar months part, the computer crashed.
The variation due to artificial light is susceptible to test - we still have primitive societies with none, and it should be possible to discover what cycles women there follow. Then there are communities of nuns, who are synchronised, live with much less artificial light, and run a longer cycle. What happens with women above the Arctic circle? The strongest influence is apparently, the presence or absence of men - research in a convent in which the researcher introduced testosterone into the air-conditioning(?), a terminated experiment, showed a speeding up of the cycle.
There is the comparison with our close primate relatives, the females of whom run an average 33 day cycle. when observed.
There is no 28 day lunar cycle, anyway. The synodic month, New Moon to New, is 29.53 days. The other cycles, such as the sidereal, in which the Moon returns to the same position against the fixed stars (27.3217 days), are mostly less than 27 and a half days. (The others are the tropical month, in which the Moon returns to the point where the celestial equator crosses the ecliptic - 27.3216 days, the anomalistic, from perigee to perigee - 27.55, and the draconic, from the node where the Moon's orbit crosses the ecliptic, either up or down, to the same node again - 27.21 days.) Only the anomalistic month and the synodic month seem to have any likelihood of influencing anything down here much, but clearly, anything running at 28 days, as a woman whose cycles are constrained by the Pill would be, will be noticeably out of step by nearly a full phase after four cycles.
I wouldn't rule out an influence. There was a time in my life when for much of a year, my cycles did seem to be irregular in such a way as to keep menstruation at around the time of Full, not New Moon. As I recall, my cycle ran short, becoming closer to a Full Moon, and then lengthened, to stay with the lunar cycle longer. This was the time that I started to suffer from migraines, which tended to happen at about the same time, on Saturdays. However, I suspect there are far more natural influences closer at hand, not least in each woman's head. It would be interesting to compare samples of women not on the pill who a) believe their cycles to be influenced by the Moon, and b) don't.
But given that the identification of menstrual and lunar cycles depends on linking a very variable phenomenon in some living systems (which originally may not have experienced them much, due to pregnancy) with a more regular cycle of a different length from that claimed, which two cycles are of similar but different frequencies, the identification seems to be more poetic than actual, and therefore very unlikely to have any importance to women travelling in space.
What could be worrying though is if the known relationship between body mass and menstruation, which leads to women who lose weight drastically ceasing to menstruate and developing other serious effects, should turn out to be between body weight and menstruation.