Lucy has too many jealous tendencies to want to share the MC with anyone let alone her sister.
I agree that she would not share the MC with anyone else, and definitely not with Charlotte.
But I think it's more low self estime/lack of confidence than jealousy.
All her life, she was in the shadow of Charlotte. Always compared to her big sister that she can't match.
Even in her love/sex life. When guys was with her, it was only to approach her sister.
She could have hated Charlotte. But no, she love her and her daughter.
It's why I find Lucy the best girl, she has a golden heart.