Yubaba (Mods of Hidden by Gods)

This is a ridiculously specific question so apologies in advance! I'm considering apping a character with previous game memories, and that game was itself an amnesia game where all memories were stripped and they had to choose new names for themselves until they remembered their lives. They've started asserting themselves as their real name since then, but their other name still holds a great deal of meaning to them. Would they be able to remember their second name if the first gets taken, or would that be too bound up in identity to keep?