I’ll say this much, though other people have touched on it…
Japanese people are VERY careful with their personal information online. If someone’s name is really unique, the kanji + reading can be used to find them. I believe that asking for her kanji could be equal to outright asking for her full name - alright in person, but definitely weird to tell someone you only know online. If you already know her full (hiragana) name, then asking for her kanji can be like asking her for her full legal name. A little sus, at the very least.
I know some adults who only use their true kanji when it’s absolutely required. They’ll use a nickname, or katakana, hiragana or even fake kanji on all SNS.
Also, sorry to be another bearer of bad news, but continuing the same conversation after knowing that it was creepy (“それ、ちょっと気持ち悪いと思う。(笑)”) could have seemed very pushy/KY, from a cultural position.
I hope you guys were able to make up though! Overcoming miscommunications like this is an important part of learning Japanese!
Good luck!