Having had the experience of belatedly discovering an estranged parent's death thanks to third party googling finding some random people on the internet talking about it... her plan means there is a good chance of post-death, unpleasant fall out for the people she loves.
Because one way or another your sister will find out, sooner or later. Not risking raising the current tensions to new heights when she finds out secondhand is in the interests of all those left behind.
I do not have mental health challenges on a level with your sister's to navigate, but still coped badly with both the complicated grief and the punch to the guts mode in which I discovered dad's death. More than a year later I still have extended periods when it becomes abundantly clear that an extended denial stage can only take you so far, and wound on wound makes it less easy to stick to the high road and not lash out with similar disinterest in other people's feelings.
Your sister sounds considerably less well placed to manage a context where a death notice can be perceived as the deliberate hurling of a barrel of salt on a badly healed wound. Personally for your own sake and that of the rest of the family I would accept that your mother will not have to live with the consequences of her choice, whereas the rest of you might well have to.
I have had to reconsider my own stance in terms of what I want in the case of my death. Prior to being on the wrong end of "don't tell" I might have been inclined to take a position like your mum's.
But not now.
I don't want my death turned into a bullet to fire at the other estranged family members, no matter the extent to which they have badly let me down. My mother invented accusations of paternal perversion with me as the object of his sexual interest. She has maintained that lie, despite the very evident cost to me, for three decades. I am 50 years old but part of me remains trapped as a terrified, horrified 16 year old being instrumentalised in a fleshcrawling act of pay back. My struggle with self-loathing and a rather flimsy attachment to the status of being alive can be traced back to the moment of her act of impulsive revenge. Which was then doubled down on for decades.
If I die before my mother, I would much prefer her not to be at my funeral and that can be organised with some planning in terms of non-announcements and less obvious choices for the disposal of my body.
But I will not let my final act in the face of what our relationship could have been, should have been, be one of vengeance, punishment, or "here's your quid pro quo" in terms of rampant dismissal of another human's needs and feelings. At the very least my final wishes involve letting her know the how, when and why of my demise before it becomes common knowledge to complete strangers on the net.
I don't want her unsuccessfully resisted temptation to take revenge, with me used as the weapon of choice, to be what ultimately defines me. But it will, if my final wishes strongly suggest that at the end of my life what she did dictated my last choices. And I don't want the family I created to have to carry my bucket of pain for me when I am gone. I want my personal bucket buried/burnt with me. As far as I am able to control the way things pan out, I don't want it to feature in any way, shape or form in the lives of the people that I love, and leave behind.
My mother's legacy is a child damaged at her own hand, for lack of impulse control. I don't want that to be my legacy too by choosing my mother's "just desserts" as the priority when I die, leaving DH and DS with both the logistics and fall out from that.
Also with my last breath I don't want to make identical choices to my father, in terms of not making room for the feelings of those left behind for the sake of making a point.
From 1984 onwards it has all been about avoiding my parents' mistakes. Will have been utterly pointless if I just go and copy their respective playbooks right at the end.
I don't think your mother's path is one that will lead to least pain for anybody. Despite the attractive qualities it promises. And grief with extra added complications can be a right bastard to manage.
In your shoes, because of my background and experiences, I would let her go on believing her wishes would be respected so she doesn't become distressed, but give myself permission to manage her passing in a way that prioritised "least harm" for everybody left behind.
And
Cos this kind of "godawful from every perspective" stuff is far from easy to have heaped on your plate.