The Loonies have vast resources? Don’t remember that, if anything they need stuff from Earth to survive
Resources probably wasn't the best choice of terms, I meant they had infrastracture; computers, rail catapults, shuttles, etc. that even modern day earth couldn't touch, much less 1960s earth. That's besides their knowledge of the sciences that are again lightyears beyond what we have even today. Further Loonies are
all significantly capable, it's brought out in it (and a few other) books that the "average civilian" in places like that has to be educated enough to perform complex math ranging from orbital mechanics to calculating how long their oxygen will last. TANSTAAFL principle also means that everybody has to be highly productive and capable to keep the colony going to every person is talented and skilled. The dangers of space weed the dumb ones out really fast.
That said, using railguns to blackmail the world can open up lots of deals
That sounds like human trafficking and I think women are expected to work in China en masse as well, they may not like losing their work force
China at the time was not modern China, but yes, sending them to the Moon would be weird and be quite similar to human trafficking. It will be quite similar and have a bad look but the Loonies still need women and if they have any leverage (and they do) they're likely to try to apply it to getting them. Loonies also have zero racial tension (can't afford it with how difficult Lunar life is) and won't care what country. Whichever nation starts sending women up first is going to have a significant economic edge over their neighbors and realpolitik is going to rear it's ugly head.
That said the moon's population is mainly male, about to be rich and powerful, and their society elevates females on a pedestal that makes modern feminists look misogynist. They may not have that much trouble recruiting volunteer women.
Loonies are likely to reserve blackmail with the railgun for an absolute last-ditch effort given their actions in
The Moon is a Harsh Mistress. Remember that the casualties of their railgun attacks were literally morons going to the areas the shells would hit, because Loonies were deliberately only firing at areas that were completely uninhabited to avoid inflicting actual damage. NASA is more than likely to gladly ship them millions of tons of poop (On a Loonie-built rocket since earth rockets can't possibly meet the demand) in exchange for spacelift capability. Hitting the Gobi desert with a giant shell to make a symbolic point probably won't come up.