The problem is as long as religions exist that say safe sex is bad and multiplying good? All you are doing is breeding more poverty. I don't know how much hate I've gotten for daring to say we should offer a one time payout of a couple grand for women to get their tubes tied and men to get snipped but the simple fact is if they'd sell their reproductive rights for a quick buck they would be shitty parents anyway and the world is better off.
But as long as you have clergy in third world countries that say things like "condoms give you AIDS" to keep people from using them? Then all you are doing when you feed the starving in the third world is breeding the next gen of beggars sadly.
You blame the religion but the true blame lies with those who mindlessly follow (any) religion without questioning both its doctrine and the men who administer it.
That, and at least around here, daring to suggest that someone who doesn't know where their next meal will come from is not in a good position to become a parent is like turning the sacred cow into cheap hamburger. It's amazing how angry and emotional some people will get when you point out what should be common sense. It's part of a larger general movement to dismantle the notion of personal responsibility and the idea that cause precedes effect or decision precedes result.
I personally would never consider impregnating a woman under such circumstances. If I somehow did, the inevitable and predictable suffering of the child would be personally my fault, the result not of "luck" or "the way things went" but my own poor judgment and lack of discipline. That, by the way, is the single number one way to become poor and to remain in poverty: having children you know you cannot afford.
So long as it is voluntary, I like your idea concerning surgical sterilization. The payout could be increased to tens of thousands of dollars and it would still be a bargain compared to what a bunch of poorly (and usually singly) parented children are going to cost society when they grow up. The only people who would truly have a complaint are those who profit from the private prison industry and politicians who need have's and have-not's to secure elections.