On my worlds of Earth, Ruquelis and Ersutiabu, there are three sexes relevant to this question, Men (XY male), Women (XX female) and Lilim (LX female), where X, Y and L are sex chromosomes. Lilim are effectively women with elongated, taloned feet and batlike wings with which they can fly (magically-assisted flight, but that magic isn't relevant to this question). Lilim- and women- children are indistinguishable until the age of 7-9 when lilim-children begin to grow wings and their feet begin to change into talons.
We therefore have crosses:
Men (XY) and Women (XX) -> 1/2 XY and 1/2 XX.
This corresponds to our familiar 50% male/female sex ratios in unions between men and women
However, we also have:
Men (XY) and Lilim (LX) -> 1/4 XL, 1/4 XX, 1/4 YL and 1/4 XY.
In these crosses, YL is immediately lethal, as the L chromosome is missing genes necessary for survival of a zygote that is only present on the X chromosome, preventing even a single round of cell division.
This leads to lilim effectively having a birth ratio of 1/3 XY Men, 1/3 XX Women and 1/3 XL Lilim, with a slightly lower birth rate than is the case for women.
In the distant past on Earth, lilim could not successfully bear XX offspring due to maternal/fetal incompatibility that caused XX woman offspring to be aborted in the first trimester, leading to a live birth ratio of 1/2 XY Men, 1/2 XL Lilim, and a significantly lower birth rate than is the case for women. Women also went extinct due to the higher survivability of lilim, until lilim were genetically engineered to eliminate the maternal/fetal incompatibility with XX offspring.
Since women reproduce more rapidly than Lilim, and breed true, this has led to undesirable social outcomes on other worlds on which populations of humans with lilim have settled. On Ruquelis, lilim prevent women from reproducing by killing or sterilizing girls who haven't begun to transform into a lilim by age 9, while on Ersutiabu, women are socially and economically discouraged from reproducing.
So, my question: How can lilim breed true once again?
I have considered putting an 'X-Polarase' gene on the L chromosome, so that during meiosis in a LX mother, any of the four resultant cells of meiosis that carry an X-chromosome will become a polar body rather than an ovum, but this would have the effect that lilim would then give birth only to lilim, as it would not be possible for them to supply the needed X-chromosome to an XY male offspring. This would lead to them becoming a parasitic sex, which is not what I have in mind.
I'd rather not go back down the road of maternal/fetal incompatibility between LX lilim mothers and XX woman offspring, so is there any other mechanism by which LX lilim and XY men can (almost) always have XY male and LX lilim offspring only?
In effect, I'm looking for a solution that maximises the LX lilim birth rate and minimising the XX woman birth rate from a LX lilim mother, rather than having fully half of their potential offspring being non-viable if at all possible. The genetic modifications need not be made to the lilim, but may instead be made to men, if necessary.
The best answer will involve the fewest gene changes and result in lilim giving birth to as few XX offspring as possible.
Additionally, since these changes must be distributed through the population by just one person who can only effect the change on one subject each ten minutes, for a maximum of 8 hours a day for a maximum of 1 month, it must propagate through the population naturally to as great an extent as possible.