I wish I had something encouraging to tell you, but my husband's past experience was similar to yours and there was no positive outcome. He couldn't afford a lawyer for a prolonged legal battle, so he agreed to giving her sole custody with him receiving visitation. It was a huge mistake, he thought it meant physical custody only and that they would primarily live with her. He had no idea it basically eliminated his rights as a parent.
A few years after the divorce, he married me, we have two children together. Meanwhile, we discovered that her new husband had been arrested twice for assaulting her. There were periods of months when my husband wasn't allowed visitation or even telephone contact with the children. We got so desperate that, after discovering the arrest, we contacted Child Protective Services just to find out if they could tell us if the kids were okay. This only enraged her further, and what little contact we had was terminated by her. We contacted a lawyer to find out what our options were and were told that we had a chance of getting at least joint custody and restoring visitation, but we would probably not get physical custody unless there was adequate proof that children were being abused. Even so, all of this would've cost thousands of dollars that we didn't have, so we were advised to "play nice" with her and try to get back into the kids lives that way until we could raise the money to go to court.. One daughter was hit by a car, we were never told. Two kids got braces that we were supposed to pay for, but had no say in when the braces were put on. (Waiting six months would have been much less of a financial burden on us.) They routinely have medical treatment that we're not consulted about. We have no input into how they live their lives. We never even get to talk to them on the phone.
As it stands now, the children have been so brainwashed by their mother that Father's Day came and went with no card and no call, same with my husband's birthday. He calls them, they don't answer. He leaves messages, they don't call back. He sends emails, birthday cards, etc. and they don't respond. The children are now teenagers with their own cell phones and, one would think, be old enough, mature enough and responsibile to get in touch with him if they wanted to. But they don't.
My advice is that if you don't have the bucks to hire a high-power attorney, then make the most of the time you have with your daughter and do everything you can not to rock the boat with her mother. Even with a lawyer, there's no guarantee that you'd win. And even if you did win, what would you lose in the fight with regard to family harmony, peace with the mother, emotional damage to your children and your marriage, etc. My marriage is hanging on by a thread, thanks to a vindictive ex-wife and her manipulation of my husband by using the children to get to us. Avoid hostility at all costs. GOOD LUCK TO YOU ALL!