We know women support migration much more than men. A feminist approach might sway them.
Men perceive migrants, who are mostly young males, as competition.
Women would consider new young men as potential partners. More potential partners can be perceived as a better choice. It naturally increases the effort of the native men in courting, due to increased competition.
For high educated people migrants are less of a problem than for people with mid or low education.
For mid education women are less of a competition than for men: A migrant doesn't have the decades of linguistic and cultural benefits to compete against an e.g. kindergarten teacher (mostly female) and can't even undercut the price (not a free market).
Compare that to a plumber.
The influx of new people increases the demand for 'female' jobs, increasing the stability of women's jobs, as the welfare state provides these services (like education).
The welfare state does not pay for e.g. plumbing. The influx of people does not necessary increase the demand for many 'male' jobs.
And since migrants are mostly male even the amount of potential competitors is far worse for men, when looking at industries, that don't have a 50/50 gender split (nearly all jobs).
For low education women welfare benefits are a bigger factor. As migrants tend to vote for welfare parties, this in turn benefits the woman.
To sum it up: Current migrants have some benefits and next to no negative effect on women.
It's reasonable for women to vote for open border and migration.
Let's use feminism dogma to solve the issue. Let's introduce a female quote on migration.
It's easy to justify ("We need to help the women, who seek a better life here as well. Right now only the evil men benefit!")
With quotas you suddenly have far more women entering the country.
Suddenly the native male population has an increased pool. I bet you could cut the amount of beta orbiters by half, by just bringing in more women into a country.
Not many will accept the neurotic mind games of entitled feminists, when literally hundreds of thousand of new women are entering the country, seeking for a stable, good relationship.
This in turn reduces the cultural and politic outreach of the (mostly feminist) women, who have learned to manipulate beta guys.
With an increased amount of (migrant) women trying to marry, more families are formed, which pushes political views towards conservatism on its own.
New women will reduce the female privileges significantly. e.g. It will be harder to find "women support" apartments, when you don't compete against 50 women in your city, but suddenly 5000. Or when your gynecologist offering free screening now has a waiting line of 20 people instead of 2.
I'm sure welfare dependent women will notice this change very quickly.
With a new, hard competition for women, native women will try to defend what they already have and will try to limit migration, to not damage themselves too much. Which reduces their will for mass migration.
tl;dr: Female quotas on migrants increases competition on women. They will try to reduce migration to protect themselves.
Use feminist dogma to increase nationalism.
To make sure: The proposal does not aim to increase migration in general, but reduce male migration, while increasing female migration to get the 50/50 split feminists like so much.