This is a great question, because it does get tricky. However, both of us agree that ultimately it’s up to your character. What name does she decide to go by?
And does she get her own memories back? Does she accept them or choose the other life she’s made for herself? Consider that the two names essentially represent her past and her present, and it’s up to her to decide which of those lives she wants.
An important part of her character growth and evolution could be shown when she’s confronted with whether or not she wants to stay with her fake name or shed it to reclaim her previous name. If you want to transition to her previous name within the narrative, this is a good and believable way to show it’s finite. Plus, it’s done of the character’s own volition.
Another important thing to consider is when your other main characters begin to see her by her previous name, if they haven’t yet already. Depending on how limited your POV(s) is/are, the other main characters might refer to her by her previous name already, or they might never see her by her previous name. If you have a general third person omniscient POV, however, you probably don’t have to worry about this.
If she does decide to take on her previous name, it’s important to show that in the story and drop the usage of her old name – since she’s essentially not that character anymore. Others may still refer to her by one name or the other, or she may occasionally slip when introducing herself, but ultimately your character will have to tell you which name she chooses and when she’s absolutely sure.
I’m glad we could be of some help! If any other followers have some advice or book recs for neko032, don’t feel shy about speaking up.