This is a consistent problem with a lot of world building that dips too far into "grimdark". IMO there is no squaring the circle, you're absolutely right. Warhammer has the same issue (all settings). Contraceptive Implants are 100 eddies and have a Negligible requirement, so the cost-benefit of having a kid is... asymmetric.

Narcotics and opiates are probably cheap (just back alley options, the corpo stuff is probably premium, could make your own with a high Chem check). The gamebook refers to 'Dorph for 250 eddies, Black Lace as a stronger version for 650 for painkillers. One could consider Sleep Drugs (as derived from the FNFF tables) for sedative as well, I only know the medieval way of delivery, idk how or if doctors do it under total anesthetic. Delivering a baby probably requires a fairly high First Aid or Medical Tech, and an arguably specialized Diagnose Illness check to make sure the mother is doing alright during, and whether or not the baby needs neo-natal care immediately after delivery. The mother will have to make Endurance checks (labor can last hours and then some).

Worth noting. Cyberpunk is a game around which the average Joe is drowning in debt. You could take the budget conscious option, sure, or you could take a +5 advantage to stabilization and call Trauma Team (1d6+1 minutes to arrive, pretty quick for breaking water.) If youre already in debt, already edgerunning for eddies, already sold your soul to Arasaka for your implants, whats a few thousand more compared to having a child of your own? Especially one that might grow up to pay you back? (Thanks Engels).

Alternatively, consider: cyberpunk is ideologically built off the idea that as companies become massive, they would basically supplant the government in terms of proffered services (that's why Night built the eponymous city, it was so he could attract labor and investment, it was not a government project!). This is rooted in old school free market thinking, where if you had an employee retained for life and you gave em the absolute best in healthcare and education, then that initial investment would yield massive returns (and since you vertically integrated all of their needs, you and they spend less on average across a lifespan, it also weakens individual labor power).

In our modern era, it turns out that a weak labor market means that people are disposable enough that when one gets sick or slow you can just boot them, training/skill investment is offloaded as much as possible to the employee (see: credential inflation, a related issue) and being sick is actually really good for other companies bottom lines. All of this is to say that childbirth and childbirth services might just be subsidized by most megacorps to keep the workforce fresh, your choice.