As you said, Rika realises later in the canon that she and Bernkastel are not actually different people. While this matters on a meta level with the creation of Bernkastel for Umineko and gameboard interactions, for Rika the human being it doesn't matter what she considers herself. She cannot create identities that way in this game because she and Hanyuu are not in control of this area nor do they have higher level access.

As shown in the canon, although Rika's memories overwrite the next Rika, it is still inherently Rika. How Rika herself comes to terms with this and copes with it as a human being is escapism, as seen with her talking in terms of the observer, putting as much distance between herself and the others as possible and crafting a different identity. As Rika, at that point, is still not a different person, we're not sure how her coping methods would matter with her name.

We apologise for the delay in response but we needed to deliberate our decision. We hope you understand and respect our decision on this matter.