G said you don't babysit your own children. You mother them. I agree. JO, when someone babysit, they don't own the children. It's a hired position.
Comment to your H, something like you don't want to feel like hired help, you are a mother and you can mother the children, but don't considered yourself hired help. You might have to tone it down a little, but not much.
The goal is to get your H to se you are the other "parent" and his "life partner" to these children. Not an emotionally frail female child care provider.
If I visited your family today with your H and kids, would I see a mother taking care of her kids or would I see H's girlfriend taking care of the bf kids?
I am not saying you should be there today and I know that is what you want. Sometimes we have to give up the old patterns we used and put on the new role, act with it (mother role) for a while so it sticks and feels comfortable.
Do you have any plans to fall back on if one or more of the kids gets on your nerves or acts out? Mothers do, usually learned from experience.