To expand on this slightly.That's as a product of society being dysgenic rather than high IQ leading to lower fertility. Historically high IQ traits are associated with high life success (still are) and high life success leads to high historic fertility; whereas now high life success tracks with low fertility. Society is dysgenic as a result. Previous eugenic materials suggested giving incentives to middle class and above families to encourage more children, and disencourage the lower classes from having children.
OP's proposed society would still likely be a dysgenic thing. IQ trends heavily towards the centre when you have children, so a man and woman who are both outlier IQ will more often have children that trend further towards the centre than they do an outlier. You wouldn't end up with loads of smart children, you'd end up with very few smart children and lots of dead children. A better solution would be to try and lower the overall birthrates of the unsccessful in society (those at the very bottom of the class system) and encourage the middle and upper classes to have children in greater numbers.
Move the centre more to the 'big brain' side than the retard side and induce a slower social change that could be more easily managed than arbitrarily wiping out huge swathes of the population. But really that's actually not important in the short term than the cultural changes that'd be needed for people to be okay with accepting birth limitations on the poor and accept that for the middle classes a large family and good home life would be the indicator of societal success (rather than just having money). I would say that the middle classes refocusing onto family life; with the move to male centred households and reduced women in the workplace that would follow would be a much, much bigger change than literally gassing parts of the population. Mainly because the sudden sharp increase in government run deathcamps would probably end the nation in its entirety; whereas a patriarchy and socially responsible populace would be much easier to maintain.