Hi, my baby girl is called Lucy Claire too!!
She is also not a fan of drinking cows milk - I have been offering it to her in a sippy cup, and she has slowly started taking a few sips.
I make her porridge in the morning with all milk, and out lots of cheese in her food etc. I also give her a big yoghurt before bed with a drink of water. So yeah I think the best way is to mix it with waterever she drinks now and just increase the amount, but if she doesn't like it then there are other ways to get her milk, or dairy into her!