That's a good and interesting question. I'd have to say that Alice really doesn't seem like a child in that world/in that text. At the start of the text, when she's playing with her cat, she is an adult's version of a child: she is overly cute, and almost arch. She's on display.
When she enters the looking glass world, Alice is several things. One is a disruption; she changes things. One is a child's fantasy. She lifts the White King up and replaces him where she thinks he should be. She is immensely powerful, like a parent to a child.
Alice is like a traveling camera eye of a person at some points. She isn't really fully fleshed out. Instead, she simply sees things that are fun for the reader to see, like the text of "Jabberwocky."