I wonder if it would go far beyond polygamy.
The best ratio they could get is 1 man per 3 women, probably worse since women wouldn't want less desirable men, maybe 1 to 4 or 1 to 5. Even then it would be an issue. How are men allocated? What's the system for who gets who? What do men even do in domestic life, watch the kids and untrap the roomba? Act as household helper for 3 women and 6 kids?
Anyway, self reliant women may not even care about being a part of a family group besides her birth family. Women might form a social co-ops from friends or coworkers, decide who's going to do child care and work part time, then the women would just go to the clinic for insemination. That way all women could get the best genetics for their kids without having to compete with other women or listen to a man saying me me me all the time. I wonder about the politics of which mothers get which genetics. If everyone got the best, people would get inbred. Could be a reason for spacer crews to meet new people. Genetic engineering could give the baby cold weather adaptations (kind if a handwave, but hey.) Really with their TL theY could synthesize what they need for biological reproduction.
Maybe men would be kept in coed cities where they could frolic and play, work out, go to finishing schools to be impeccably trained to be pleasing to women, get jobs in the music, media, and sports industries, and generally be trained to serve as the masculine archetypes that the women want. It could even be formulaic, like x% of men will be trained to be fun loving partiers, x% extreme sports competitors, x% brooding bad boy musicians, or whatever career/archetypes market research says will be most pleasing to women. Wider female society could have idealized media stars follow instead of real people they'd have to compromise to have a relationship with. The whole thing could be a cynical psyop to give women thrilling media to keep them on the job, give men something to do so they don't become a law enforcement problem, and to keep the current elite in power.
And creepier!
If their biotech is as good as you suggest--and it might be-- why haven't they just fixed the sex imbalance?
This gets back to your questions about why the female domination of males was maintained for centuries after the revolution.