Well, I finally managed to figure out how I'd design the Red Queen. I did get some inspiration for Snow White's Evil Queen but I also got inspiration from a version of the Red Queen that came out in like the 80s or 90s, I forget. She's a character that is often confused as the Queen of Hearts but is actually a completely different character. She's bossy and overbearing but is actually helpful to Alice by explaining the rules of the game and how she can become a queen. She is TECHNICALLY an antagonist because of how her and Alice are on opposite sides but she's not really an antagonist because she didn't try to hurt Alice and is always civil when Alice is around. She also likes to keep violence to a minimum, unlike the Queen of Hearts, who wants to behead someone for every little thing that upsets her, and prefers to help events unfold as smoothly as possible. While she's still crazy like everyone in Wonderland, she is the sanest out of all the residents of Wonderland.

You might not notice but her crown has jewels that are very symbolic because they're chess pieces:
The middle one on the very top of the crown is a Queen chess piece
The middle one below is a King chess piece
The jewels next to the Queen are the Bishops
The jewels next to the Bishops are the Pawns
The jewels next to the King are the Knights
The jewels next to the Knights are the Rooks