Firstly, yes the shinki would accept staying at the temple.
Chances are good that if Zelda finished introducing her to a lot of gods and then asked if there's anyone she'd feel more comfortable with, the shinki would say no. On introduction round 2, if she gets the same question, the shinki would actually ask something to the effect like "I thought you were testing me before you named me, but if you're trying to get me to leave, I'll leave."
I'm torn because I can't see any reason for her to want to leave Zelda considering everything you guys did in the Second Fight thread, but I also know you were trying not to make another NPC for us. So I think it's ultimately up to you if you want Zelda to name her in the end and how we go from here.
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Chances are good that if Zelda finished introducing her to a lot of gods and then asked if there's anyone she'd feel more comfortable with, the shinki would say no. On introduction round 2, if she gets the same question, the shinki would actually ask something to the effect like "I thought you were testing me before you named me, but if you're trying to get me to leave, I'll leave."
I'm torn because I can't see any reason for her to want to leave Zelda considering everything you guys did in the Second Fight thread, but I also know you were trying not to make another NPC for us. So I think it's ultimately up to you if you want Zelda to name her in the end and how we go from here.