Fascinating post Tortoise. Much of the reason that I am following this case is because I'm trying to work out JG's psychology.
What you have posted is exactly what I gleaned that JG is like, and what I guessed that her relationship with her children was all about.
From what I had read, I had thought that she was juvenile. Her searching for spells to make BB love her, and writing silly prayers to goddesses are the type of things that an adolescent girl would do.
I had wondered what on earth JG sees in BB, but it makes sense if JG's emotional and romantic development is stuck in the adolescent stage. Adolescent girls often develop raging crushes on the baddest, most confident, most dangerous, most aggressive boys, mistaking that behaviour for being alpha males. It makes the young girls feel special and flattered to imagine that while their crush is dangerous to everyone else, he is kind to her because he loves her so much.
Fortunately, as most girls finish puberty, they realise that most of these dangerous boys are worthless losers. Also, the way girls become attracted to boys matures: adolescents develop fast, spontaneous, addictive, intense, irrational, uncontrollable, hormonal attractions, whereas adults slowly grow to like the objects of their desires, taking into account more important things like personality, compatibility, shared values etc. Adults have more control over who they are attracted to.
It seems to me that JG never matured past the adolescent stage of developing addictive crushes (that she mistakes for love) to "bad boys".
Also her googling about being addicted to a man who doesn't love her (presumably a different man to BB since it was before they met) shows her mindset. Love chemicals can be addictive; if we view JG as an addict and BB as her fix, her behaviour starts to make sense. Many addicts will neglect or sacrifice everything and everyone else in their lives, just so they can get their fix.
I agree with your impression that BB loathed her, but would charm her with glimmers of care and affection to keep her clinging on. He would beat her right down (literally and metaphorically) then uplift her just enough to keep her hopes up. That is a common dynamic with abusive partnerships.
I also agree that JG viewed her children as little more than props in her fantasy family life with BB. She got pregnant immediately upon meeting him because she childishly thought it would trap him and make him love her, then repeatedly got pregnant with similar intentions. When the children were with them, she viewed them through the lens of her fantasy relationship with BB. I believe she only fought to get them back so that she could play happy families with BB, to feed her fantasy of love and life with him. I believe BB only fought up get the children back to prove his innocence, and out of a sense of possessiveness. I think them working hard together to fight for custody of the children would have fed her fantasy of them having loving unity together.
In her prayer writings, her hopes for her children are mentioned near the end almost as an afterthought. In statement analysis, order indicates priority, so her children were low down in her list of priorities.
I also think that she may have sometimes bullied Ellie to impress BB and show solidarity with him. Like in the texts when BB called Ellie a **** for wetting herself, and JG agrees that she should be tougher on her and that she wants BB to be tougher too.
I suspect that JG's standing by her man is partly because he is her addiction, and partly because BB will throw JG under the bus and tell the court of any abuse that JG participated in if JG doesn't support him.