In order to prevent Merrill from doing it. She did it out of love for Merrill. She saw which way the wind was blowing and couldn't stand to allow Merrill to do it herself. She sacrificed herself for Merrill. It's called a sacrifice because it means that there's no favorable outcome. Her choices were: wait around until Merrill succumbs to the Pride demon (it's very telling that Merrill was too prideful to admit that she wasn't in full control) and have no control over the when/how of the possession -OR- take the bullet herself, thus saving Merrill and having some control over the when/how of the possession and allowing Hawke to finish her off.
Is there an element of 'pride' in Marethari's actions as well? Sure. She assumes that she can handle it better than Merrill can (and, in my opinion, she's right). But all she did was speed up the inevitable. Audacity had found a willing, weak 'victim' and Marethari did what she could to try to minimize the loss that was inevitable at that point.
That's how I read the situation.
I'd argue the love bit, or at least that it's not the best type of love out there. I mean it's love, but it's.... warped in expression. Marethari's always been a person who falls into the My Beloved Smother trope, as she kept Merrill from having a healthy social life within her own clan by forcing her to study since she was a child. Her pride makes her think that only she can possibly be right, because she is the Keeper. No one else could possibly have a point on a matter that contests Marethari's (limited) viewpoint.
She's a Keeper, she's a Mage, she's a mother figure, and she's old as hell. She's prideful as all hell. Indeed, she keeps the clan around not because they need to be there, but because she wants to be there (it's to the point where some members are willing to go the route Velanna's "clan" went when she was forcibly exiled).
Demons are drawn to those with power, literal and figurative. Marethari has both. She has the powers of a Mage, the training of one, and influence over not just her clan but others as well. Throw in the divide between the two respective Mages and there's ample reason to suspect that Audacity was after the Keeper all along (as Merrill has only being a Mage as her schtick) but it's also possible that he really didn't care by that point.
There's a third option: Go the frick to Kirkwall and bloody talk about this with Hawke and Merrill while also having your clan on alert. Hell there are more options then that. Saying there are only two options falls into the false dichotomy that plagues a lot of things. You always have options at your disposal, not merely A and B.
But Marethari didn't even want to study the Eluvians (from DAO and DAII) so her pulling information out of her ass doesn't fly.
Your argument would have merit had she jumped right to her final action. But she didn't. She tried to reason with Merrill. She tried to bargain with Merrill. She tried to enlist others to convince Merrill when she saw she wasn't being successful. As an extreme action, she banished Merrill in order to protect her and the rest of the clan. None of this worked. Merrill was too prideful to turn off of her path and Marethari saw no other options after expending effort in so many other ways to convince her.
I'd hardly call "having members of the clan revile her" enlisting others to convince her, particularly if lies are fed to them to turn them all against her. And if you mean Hawke, well she did break her deal with Merrill. The deal was that if the Varterral was killed the item would be given to Merrill. Hawke is merely a witness to the arrangement.
And she didn't banish Merrill, Merrill exiled herself. Marethari will tell you that she's not the one sending Merrill away, Merrill's doing it of her own free will (something Merrill says as well).
Now... hopefully this will be my actual going to bed moment.