No, as you point out later in the OP you’re technically “daddy’s girlfriend” not her stepmother, and you probably shouldn’t expect that to change unless you get married.
Why are you stuck with her all weekend? If you’re boyfriend doesn’t want to watch his kid, they’ve made it clear that you’re just “daddy’s girlfriend”, then it shouldn’t be up to you to watch him. Tell him that you’re happy to spend time with his daughter but he’s going to have to be around for that to happen.
Marc