Caveat: I'm neither a woman nor a doctor.
The plausible mechanism for LC/VLC working to help PCOS is that it induces ketosis and a more fat-burning metabolic orientation. The weight loss engendered (and the burning of excess adipose when fasting, especially) decreases insulin resistance (which feeds into the hormonal cycle). Adipose tissue also tends to aromatize androgens to estrogens, which further inhibit FSH. The increase in LH maintains the unopposed estrogen dominance, creating insulin resistance and hyperglycemia ad infinitum.
Weight loss is one of the primary interventions begun for PCOS in all settings. Since LC/VLC is almost universally effective for loss of excess adipose, it can only help in terms of getting one to a better place. If one could maintain a healthy weight without being LC/VLC, it probably wouldn't hurt to increase the carb intake moderately. Probably of equal importance is the 3:6 balance, which will normalize the pulsatile hypothalamic GnRH secretion that regulates LH/FSH cycling (and thus the estrogen/progesterone cycling that is pathologic in PCOS). Plus, progesterone is easier to make from cholesterol than estrogen, so replacing carb intake with fat intake (including cholesterol intake) should also help the process.
Not exactly what you were asking for, but when I didn't see the pathophysiology begin to appear, I thought it might be useful to throw it out there (albeit very roughly sketched). And at least I got to use the caps-lock a whole lot without looking like I was screaming c: