My dd was trying to do the same thing, so I started giving her a bottle when she woke up in the middle of the night. I know I'm not supposed to, but she'll take 7-8oz that time so its worth it to me. I just brush her gums afterwards and put her back in bed. I get less sleep, but I know that she's getting enough calcium.
I also wanted to add-if she prefers an open cup, that's great! Why not encourage that instead of a sippy cup? My dd does great drinking from an open cup.
Good luck!
