Everyone is making a big deal out of a "problem" that is ridiculously simple to solve:
Every government on Earth says to its citizens: "You can either get this treatment that will make you live this much longer... OR you can have children... but you CANNOT have both." Have a police force charged with enforcing this rule, and nothing more. Make the penalty for disobeying EXTREMELY harsh... like forced abortion, or the child can live but both parents are put to death right after.
Sound harsh? Nope, it's a choice for every single person, nobody is forced into anything.
Think it won't work? It ABSOLUTELY will, and here's why: humans, the VAST majority, are at their core selfish beings. Given the choice between living hundreds of years, or maybe even indefinitely, or having children, MOST of them are going to choose the selfish option and say they'll give up having kids. Guaranteed.
Now, before you say it, this plan needs some tweaks to cover all the bases... for example, there's got to be some minimum age where this decision is made. Is is 18? 21? You want someone to be old enough to make a proper decision of course... but then, what do you do with anyone who has a kid BEFORE that age? I'd suggest maybe a stiff fine, stiff enough to dissuade MOST people from having a kid before the designated age, but you're still going to have some and we don't want to be killing teens, right? But, teen pregnancy has been on the decline anyway, so it seems that education works, so it's probably not a dealbreaker.
Do we perhaps make people have to undergo a procedure to render them unable to have children when they decide? Would certainly make he policing requirement less, but that might be a bit much... worth debating at least.
Of course, you still have to ensure enough people do procreate to keep the species genetically viable and not stagnate... maybe a lottery? Half the population gets to live forever, half have to procreate, and it's all random? Hmm, lots of chances for abuse of course, and that aside, it kinda hurts my "democracy" brain that says everyone should have an equal chance at the good life... maybe just limit it? Like, if you get picked to have kids you can only have one, and you only get to live an extra 200 years or something? That might seem like a good trade-off right now, but when the average lifespan is 1,000 years it's not going to seem like such a good deal.
So yeah, I'm not saying there aren't some holes that have to be plugged to make it all work right, but the point is if you simply give people the choice between children and a massive increase in their own lifespan, the basic problem is going to solve itself based on nothing but simple human nature, that's the bottom line.