That's not the way you control birth rates.
You control birth rates by controlling FEMALE fertility... not male fertility.
There are mathematical reasons behind this, but for the purposes of simplicity I'll give the lay explanation. ie - A female can only have one (or so) child every 9 months. A single human male can produce one every day. So having a population with only one male, but 100 females, allows for a FAR higher reproduction rate than a population with 100 males, but only one female.
I'm only saying that you ascribe certain properties to the populations you are dealing with. And there are definitely places in the world where your assumptions fail to hold up. But biology ALWAYS holds up.
By the way... I was being kind. Rape is only ONE whole in your plan. There are many. For instance, your plan also assumes women will never simply get horny. Or that they will only "get off" in certain predetermined ways that your plan deems acceptable. Your plan assumes a lot ... and all of it from your own perspective and belief system. If we work within biological limits... we achieve our goals regardless of the local belief system.
This is the point I'm making... you are asking that human females satisfy themselves with that which YOU deem it acceptable to provide them. What happens if a human female, indeed many human females, decide they want something different???
But it's a well documented effect. Seems to me the easiest way to cut reproduction is to simply develop countries.