>>123086neat post with perspective (esp cults and false imprisonment), even if I disagree with some details
I don't think at this point OP is able to do much convincing towards her sisters trannyism or whatever, but I want to theorize on this a bit anyway.
You can't reason someone out of a problem they did not reason themselves into. Or at least not being aware why that person believes what they want to believe.
In this case, OP's sister believes becoming ftm will give her validation - notice how trannies and allies try to empathize with each other no matter what, something abusive homes never give.
It will also give her a way out from misogyny - same reason girls may become nonbinary. She might not even want to transition for all we know but instead she just likes the fantasy of escaping being looked down upon, forced to live a certain way etc.
Finally her only friends or people she talks to may be trannies or some lgbt thing and that obviously changes how people see the world - in fact if she got chuddy friends she may change her mind.
Basically people become trannies as a solution to emotional problem IMO.
I don't think unnecessarily fear or hate mongering is a good solution because that's just creating another emotional problem, but it may be a good idea towards someone who is already causing her grief like an abusive dad/authority figure or something. That could work well yeah.