OK, so "The Birthmark" is tricky in that Aylmer thinks he knows what the monster is from the start and then gets a rude awakening at the end. From Aylmer's perspective, the monster is Georgiana's birthmark and all that it symbolizes: human imperfection.
It seems as though removing Georgiana's birthmark will be easy as a Botox shot. Aylmer is eager to perform the procedure on Georgiana, and she agrees to it.
All was good until Georgiana – and the reader – discover that Aylmer was faking the cool exterior. The procedure won't be all that simple. She's frustrated that he lied to her, and he's frustrated that she left the "magic circle" in which he tried to trap her (i.e., the boudoir).
Mmm… that's about as bad as it gets.
This is where we break from the standard "Overcoming the Monster" plot. The monster, as it turns out, is not Georgiana's birthmark, nor human imperfection. Instead, it's Aylmer's obsession with making a necessarily flawed mortal into something perfect. Ah, the old Monster-Switcheroo plot twist (not actually a typical plot twist).