"Where do I stand as a father?"
Leaving aside the existential elements of this question, I would answer this based on your obligations.
- Moral - your daughter must be safe first and foremost. This should take precedence even if in conflict with your legal obligations.
If you believe that your daughter is physically or verbally unsafe in your ex's home, then you cannot allow her to return. If you go down this road, you will be going against a judges legally binding order and you should immediately seek to set your relationship to those obligations on the right footing (see below).
Simply not having a good relationship with your ex's family will probably not sway a judge, especially given the numerous criteria a judge will use to make a custody decision.
Where the line falls between any mental/verbal abuse and just not getting along would be much more challenging to parse out, and if you're not clear on this, then seeking professional guidance on this issue would be in order.
- Legal - if you have shared custody defined by the courts, you have a legal obligation to abide.
In the US, some states allow or require a judge to consider the child's preference and in some cases allow the child to speak in court about that preference, but it will always be the judges decision. Georgia stands alone in allowing a 14 yr old to choose as long as the judge does not disagree.
If you want a different situation and cannot come to a mutually agreed upon arrangement with the mother (the best solution in my opinion), then you will have to appeal to the courts to change the arrangement and take steps to ensure the presiding judge is aware of your daughter's preference.
- Dad - you have to be there for her.
As a father of three girls who split time between their mom's and my house, I have an obligation to be clear with them about the realities we are living with. They understand what our households have to do live by the legal obligations that separation has created.
Further and personally, I do the hard work of maintaining a relationship with my ex that works best for my children, not me. For my situation, this means bearing, without rancor or resentment, responsibilities which are not fairly divided. This is a purchase, a transaction, that I pay for two reasons. Firstly, because it's the path to peace for both our households. The burden of their parent's split should not be borne by my kids. Secondly because if I carry anger toward their mom, then I steal peace and contentment from my children and my home.