It's none of your mother's business. You don't need her permission or approval. You don't need to run your procreation plans past her, no matter what you decide to do.
Likewise, your mother is entitled to her own feelings. Whether they're right or wrong feelings is a matter of perspective. You don't have to LIKE how she feels about your choice but you do need to RESPECT that she may feel a certain way about it ... just like she doesn't have to like your choice but she needs to respect that it's YOUR CHOICE.
You're an adult. You need to learn that it's not your responsibility to get people to "side with [you]." Your job as an adult is to make your own decisions and then simply say "Sorry you feel that way" to the naysayers. Stop worrying about gaining Mommy's approval (especially since it seems like you may not get it) and instead make peace with the fact that you just need to come to a mutual understanding to butt out of each other's feelings and choices.
If you're not harming yourself or anyone else, then everyone else needs to butt out of your life. Of course, most people will NOT butt out of your life, so you need to learn how to ignore them, and not waste your time begging them to see it from your point of view. People are allowed to have their own opinions, and it's out of line for you to think that you are allowed to convince your mother that sterilization is a great thing - just like it's out of line for your mother to demand that you give birth. This is why you need to learn to limit what you tell people, because (a) not everyone gives a shiiiiit what you choose to do with your life and therefore doesn't want to hear it anyway, and (b) when you reveal personal information about your life it's practically an invitation for others to comment on it. So don't reveal personal info about yourself to people who don't need to know about it, if you don't want them commenting on it.
All that being said - if you are on her health insurance and/or living under her roof, then she's going to expect a say in your life choices. That's the catch to depending on someone else to support yourself - they're entitled to weigh in on your choices if they are helping to fund or support them.
So you need to think long and hard about whether you're willing to risk pissing her off in order to make this choice ... are you prepared to deal with an argument or a huge fight? Can you support yourself if she removes you from her health insurance in retaliation, and/or throws you out of her house? Is this a choice that's better suited for the point when you can fully support yourself and therefore you don't owe it to her to even inform her of your actions in the first place? Or if you feel that this is an urgent matter and needs to be done ASAP, are you willing to deal with the repercussions?
Your choices are your own. Her feelings are her own. Normally I would say to not even inform her of what you're doing because it's none of her business ... but if you're depending on HER insurance to get this procedure done, and if she's going to wind up finding out about it, then you need to learn to deal with the aftermath.
Have you gotten therapy because of your abuse? What about for your need to gain Mommy's approval? I would personally suggest that you pursue THAT before you worry about a life-altering surgical procedure, frankly.
And that's assuming you are even able to GET the procedure. Right or wrong, there are many doctors out there who won't do it for someone as young as you. And if your mom has "Catholic insurance" then they probably won't cover it either.