I don't know if LH was/is suffering from battered wife syndrome or not. There's not enough information right now for me to entertain that possibility.
Based on the info we do have, it seems to me that LH has no fond memories of middle/high school (at the funeral, she stated that she's happy Cooper won't have to experience those years), when LE gave her the tragic news that her baby was deceased, she didn't ask to see him - she only asked to see her husband, and has stood by her man throughout this terrible ordeal.
IMO, LH is bitter about her teen years because she may not have been "popular". She may have pined after a boy who didn't notice her. She may have felt like an outsider (her funeral statement about "who to sit with at lunch" suggests the possibility that she was shy and may have had difficulty making friends during her middle school years).
Fast forward to when she met RH. She may have finally gotten what she always wanted but never had: the attention of a man.
The above is pure speculation, but I have no doubt that the most important thing to LH is having a man in her life. It's possible that Cooper was never as important to her as having a husband was/is, despite years of trying to get pregnant. She may have naively believed that getting married and having a child would bring her the utmost fulfillment. She may have become disillusioned with the reality of the challenges of marriage & parenthood versus the fantasy of happily ever after.
I don't know if she's involved or not in her baby's death, but some of her statements strongly suggest that she may be, IMO.
One thing I'm absolutely convinced of is that her top priority is her man, and I find this beyond reprehensible given the circumstances. I also find it disturbing that she expressed a desire to have more children with the man who has been charged with killing her baby - a man who has only expressed concern for himself and has only bemoaned the unfairness of what HE is going through, all the while oblivious to the horrific death he caused their baby to suffer.
I cannot understand how a woman can stand by a man like that, unless she's either as self-absorbed as he is, or is also complicit in their baby's murder, or both.