Incentives for sterilization?
You wouldn't have to sterilize portions of the population forcibly: you could offer incentives (available to all but only significantly attractive to some). A couple of examples that come quickly to mind: licencing (and perhaps providing free of charge to the user) "recreational" drugs to anyone who has been sterilized; or paying them an extra benefit, perhaps 1.2 x one child's-worth of child benefit, to cover that parent's part of the 2.4 children of a "standard" family. Or maybe both; they'll both appeal to those lines of descent who expect to do forms of work that have now disappeared, and that raise people to think of bettering themselves as betraying their class.
But that would be long-term thinking (i.e. beyond the next election), and so not of interest to your average politician.