The rich having extra children is not a problem. In fact, it's part of the solution to wealth inequality.
First, more kids means cutting the pie into more pieces. If that means more education costs, that's fine: we get more educated people, and through much of history, people from the privileged classes were the only ones getting higher education at all. We still managed to make some headway then, so this isn't a show stopper. It's far from ideal, but it's not going to destroy society, especially if it just means the proportions remain skewed toward the upper class rather than being entirely made up of them.
Second, when the parents finally do die, the fortune will be distributed among more children. This has historically been shown to break up empires.
Third, even if the top 1% has ten kids apiece (which they won't), while everyone else has just two, that results in them becoming the top 5%. This will increase the population far less than the other 99% finding ways to cheat, even if the 99% cheats less often per capita.
Fourth, this is a treatment of aging, which could arguably be classified as a disease in its own right. It is not immortality. If that means productive and high-quality life span is greatly increased, but total lifespan is still limited by other factors, then all we have to lose is the concept of societally funded retirement -- a notion some of us already see as impossible for ourselves, and that in many places never caught on in the first place.