I'm sure you have plenty of great answers here from these mammas.
This is a hard situation. Love, Love, Love. Just keep loving her and all of them. Be kind and good and interested. You're doing great.
Maybe you can have Lilli more often, it sounds like that is what she needs and that's okay. She's the only girl and that makes a difference too. Give her what she needs now and less problems later, especially as a teen. And so what does she need? Security, Stability, Love, Understanding, Warmth, Joy, Smiles, Laughter, A Sense of Belonging.
If you're willing and able to take care of her and the others, just do it in joy and tell this to the sm. Obviously this child wants to be with you, let her be. Also, children do not take well to transitions, even in a "normal" setting, and so address that by telling her what is going to happen and make it light and happy. Sing a song, the same song when it's time to leave and when it's time to come in. This makes all the difference even if it doesn't seem so, especially at first. ~~ When it's time for the children to go home at our house we sing the following in a very calm, sweet way: ~(we started this when they were about 2 but you can begin now and keep it going, it didn't always work to save the upset, but it still brings a sense of transition and order and love that children long for) ~ "Good bye, good bye, we traveled so far from heaven above we shine like a star."
When we enter the house or when we enter a new situation, we take a moment and quietly say as we touch our head, lips and heart: "Clear thoughts, wise words, kind heart." Then we sing with matching movements: Where is thumbkin.
These things alone when done in calm, kind, gentle, light ways consistently helps greatly. It's something everyone should do. We must allow ourselves to enter the child's world, their innocence.
This family which you are a big part of needs to find a way to blend as a working, loving unit, not to just do the motions of every day stuff but to find the meaning and life in it all. Perhaps you just need to take her more and everyone be okay with that.
This little sweetie needs you as I'm sure the boys do too. But maybe, if it's feasible, she needs to be with you, live with you. Is that a crazy thought? Maybe not.
My prayers are sent for you and Lilli and all the little ones and adults for joy, for doing what is truly meant to be for each and all. LIVE and LOVE