I am at a conflict. Already 1/3 way into my story, I have two decisions to make. First off my main character has 2 names. Her birth name and her given/earth identity. I want the character to know that her name is not the name she is known for. How from here do I continue to use her given name or her birth name. This is being written in third person and the MC accepts that her given name doesn't seem to suit her.
I think it should depend on who is using it. Assuming it's clear to the reader why the name changed, use the name people use who are referring to her. If it's the narrator, you may want to either introduce the matter sooner and stick to one name even though others call her something else. Or have her use the old name, then remember that isn't her name anymore. The idea is to fit it into the story, that's how to decide.
Id just finish your first draft and go back and change the name in your editing/rewriting phase. Going back now and changing it will detract from your momentum and might start an endless loop of editing before you're even finished.