Oh, I am so sorry. I know exactly what you are going through. One of my son's would do exactly the same thing. It would be so bad that my older kids started a mean little game just because it was so funny. Humor was one way I tried to deal with it. One of them would use one finger and touch me on my shoulder and the baby would scream and scream. I mean that one example is a bit backwards from yours but it just reminded me of it.
All i can suggest to you is to maybe give her some responsibility for the baby. Tell her that your responsibility is to take care of her and someone needs to care for the baby and that leaves her. Which means, either holding her, laying on the floor playing toys with her, "reading" her a story, giving her a puppet show, feeding her a bottle, etc. Put some of the responsibility on her and see if that helps. Have her check to see if the baby needs to be changed yada yada. And when the baby sleeps, then you can promise her a reward for doing a great job taking care of the baby like baking cookies (nothing complicated so you don't get stressed just get those cut and bake ones), playing playdough, reading, tickling, watching a movie, whatever. And if she doesn't do a good job with the baby then well, she doesn't get to do something "extra special with mommy" she has to play by herself when the baby sleeps, or watch a movie by herself, or take a nap or whatever. Nothing hateful or punishment, but nothing special either.
Hope it helps.