And if the reflection is in the hospital and pretending to be Valkyrie while talking to the real Valkyrie on the phone, it makes no sense for Valkyrie to start being possessive and telling the reflection to not call her dad "Dad".
If the reflection starts telling the story about what happened and saying "your father" while she's in a public place, people might put two and two together, you know? "Gee, it sure sounds like she's telling the story of what her dad did today, but she's calling him 'your father' instead of 'dad'. I wonder if he's her real dad at all?"
Or what if her dad or mom or a friend comes by and hears her talking? How'd she explain obviously talking about her own father but calling him "your father"? How'd she explain who she's talking to?