This idea that people with eternal youth would be constantly reproducing is not necessarily accurate; nor is the idea that producing any more than 2 children per couple going to result in population growth. Additionally, China's one child policy has had massive repercussions for the State that are only just now starting to bite.
In reality, population sizes in an enlightened immortal community would tend to police itself, but let's break down why this is the case...
Let's start with reproduction itself. Conceiving the child is fun; raising it for 20 years? Not so much. That's not to say that it's not a rewarding and fulfilling part of your life that you don't look back on with pride and satisfaction, it's just that raising children is hard work, especially in a developed society. It's also damned expensive in that same society.
So, couples who want to raise children will raise a few children, then when those children go off and live their own lives will use their eternal youth to pursue their own interests. Best of both worlds. In today's world, juggling a family, work and personal pursuits is very hard because we have a set amount of time in which to do all these things; that means we have to do them in parallel. Often people choose what they want to do by priority, and discard the lowest priority live choices because they want to focus on what they really care about. In many cases, this means people choose NOT to have children.
Of course, the other reason why some people choose to have children at a given time in their life is their biological clock. They don't know if they want to or not, but time is running out, so to speak. So they take the plunge, 'just in case'.
In an immortal society, the reverse in both cases would be true. Some people would choose to have kids because they can pursue interests largely in serial rather than parallel, and devote a certain amount of time to children, after/before which they pursue other interests. There would also be less impulse choices around children as eternal youth means that there is no 'biological clock' to force a decision at some point.
All that said, some people will still choose not to have children, or will be unable to have children. What that means is that those having children have to have more than 2 to maintain the population, let alone for it to grow. If only half of the population pairs up AND wants children AND can actually have them, then every breeding couple needs to produce 4 children in their lifetime to maintain the existing population size.
To use China as a case study, they told every couple that could and wanted to have kids that they could only produce 1 child. What was the result? Runaway ageing population. China has a serious health and aged care issue today and it's getting acutely worse, where it's predicted that in the next 10-20 years there could be one old person in need of care for every 3 working age people in the country. That's not a healthy ratio.
By comparison, the old age pension in Australia was enacted when there was 1 retiree for ~60 workers; that ratio is decreasing rapidly and as such even Australia is rethinking how we define retirement and social welfare for the elderly, not because of heartlessness, but because of sustainability.
Also, your question assumes that eternal youth doesn't impact a woman's fertility or that it perfectly restores her post-natal body to its pre-natal form. Neither of these assumptions are probable.
The concept of eternal youth is usually predicated on one of two approaches:
- Slowing down metabolism and cell replication
- Increasing the effectiveness of cell repair and replication
If you consider the first approach, this means that not only is a woman's body unlikely to recover quickly or well from a pregnancy, but the release of eggs for fertilisation is also likely to be slowed. This means that women would be less fertile in direct proportionality to the increased lifespan. There is also a chance that gestation would take proportionally as long - women who now live 800 years instead of 80 might take around 8 years to bring a baby to term. It also means that women would leave children to the end of their new lifespan because of the impact it would have on their body.
Option 2? Well, increased (even aggressive) cell repair may well impede the female body's ability to maintain a pregnancy as this generates many changes for the woman during the baby coming to term. It may well lead to many miscarriages and the metabolic requirements would mean a massive increase in food intake which would also put massive pressures on the body to process and supply to the baby as well as the mother.
Finally, we're also assuming that children will grow and leave the nest (so to speak) at the same rate in this environment. Assume children grow and mature physically at a normal rate until they're 'given the treatment'; they'll be physically mature by 20 like normal. The problem is, they won't be financially, intellectually or even emotionally mature enough to survive in this culture by themselves. They'll be competing with people who've been productive for centuries, building wealth, experience, skills; it's unlikely they'll be leaving home for decades to come. That alone could inhibit parents from wanting even more children.
Put simply, eternally young mothers would probably have a proportionally higher energy or time cost to reproduction in the first place, limiting childbearing to lifetime numbers similar to what we have now. They may well stay at home longer as well, meaning that people are less willing to take on such a commitment, and even if none of this is true, we still need a critical threshold of more than 2 children per breeding couple to replenish the ranks before we worry about runaway population growth.
Even if all that fails, the best possible thing you can do to limit population increase is education and gender equality. Take a look at the birth rate statistics in developed nations where there is a high degree of education and women participating in the workforce; it's close to zero. In the case of Japan, it's actually decreasing. Australia (for instance) is growing in population ONLY because of immigration. We're not having children at the rate we used do and are barely sustaining existing population counts via that method.
All things considered, while it's important to consider this issue before you release your treatment to the populace, I seriously doubt it's that big an issue for you, especially if your society promotes the value of education and equal opportunities for all. If it does, then your population is going to be too busy being great at what it does to be creating a runaway population size problem to solve.