My girlfriend's Ex is threatening to keep the kid from her.

I recently started dating a wonderful, loving woman who after years of being a stay at home mom is working hard to becoming a nurse. She currently works at a hospital in the Sierras. Her ex (3 years divorced) is a very controlling man and the second he heard she was seeing someone else he flipped and is using the 6 year old as leverage. They have joint custody of a beautiful daughter. The ex husband lives in L.A and the girl lives with him. My girlfriend flies down once a week to visit her daughter. She spends around $500 per trip and wants to bring the child to the Sierras half of the time. The main reason for this is she is very close to her daughter and being away from her breaks her heart, but the secondary reason is money. Being a stay at home mom she wasn't in the job market and her husband was which gave him an advantage when hiring a great attorney. As it stands the child cannot leave the county without the father's consent and he looks like he's going to make things quite challenging. Any and all advice is welcome. I have strong feelings for this woman and can only help so much even though I wish I could do more. Thanks in advance.