My plan would be to influence young men to find a partner, commit and settle down as young as possible, with clear messaging that this is a route to happiness, as well as social wellbeing.
Indicator : more committed couples having their first child before they turn 25
I mean this is all well and good but most couples under 25 don't have secure housing, don't work in well-paid jobs, and don't have a ton of savings. The absence of these things, whilst not determinative of happiness, makes happiness more difficult. Add an incredibly expensive tiny human being or 3 into the mix, and bam - instant poverty and stress.
So what you're essentially saying is that you want to tell people that having children before 25 is the route to happiness, knowing it isn't true, and so gaslight UK citizens into increasing the fertility rate?
(I realise you've said improving housing is something you'd do alongside this but that doesn't solve the absence of well-paid job and savings issue.)