Hi there, I don't think it's because of her wanting to be in your bed. They don't have object permanence at her age so she can't actually visualize you being out there in your bed and wanting to be with you. When you aren't near her, you don't exist to her. In a couple of months that will change :-)
However, if you help her to get to sleep with AP then that can become a prop and she will not be able to get herself back to sleep without it and that may cause her to cry for that prop. All she knows is she's awake, doesn't want to be and wahhhhh!
Have you read about shush/pat in Tracy's books? Is your LO still swaddled and what are her room conditions like? Does she take a paci? If you know for sure she's not hungry, given her GS, then I would suggest starting with shush/pat to resettle her...then she'll learn to get herself back to sleep in the night.
4 mths is when the world starts to really stimulate them. How does she go down for her naps? Her first nap looks a tad UT and I wonder if maybe she's ready to increase that AM A time towards a 4 hr EASY?