I think this is a really difficult one, to be honest, because you want your DD to have a relationship with her father and spend time with him, but in a way which is positive and does not leave her distressed. However, your own experience of her father is a relationship which left you distressed to the point you left and that is still very raw. So you need to separate those two things out in your mind.
There is a lot of discussion on this board about what courts would do, but the fact is you do not know what a court would do until you are there, and the child’s wishes should be taken into account as well as any impact on the child. There are lots of steps along the way to court to try to reach agreement about what the child’s best interests are. It is best that you can both work to reach a plan which your DD is happy with. That may take time. It might be easier to look at mediation as a process to try and reach agreement rather than you will sort the issues in an hour. Make sure you have good real life support in this process.
If I were you, I would be going into mediation saying DD cannot manage the contact plan, because it is too different to what she has been used to, and seeking to review it in a manner which meets her needs. Yes, a child needs a relationship with both parents and you are not saying anything against that, but what you ideally want is to be able to work with her father to ensure that time with both parents works for DD and you can both take steps to help her build her relationship with her father. That will be better in the long term for her. Yes it grates and is stressful if there has been emotional abuse and he has been an absent parent, but you need to separate out his relationship and hers with him.
The fact that she has grown up with don’t upset daddy might mean that she does not feel relaxed and comfortable when she is there, so the question is what to do to help her feel more comfortable.
Personally I would take ‘what a court would do’ out of the picture because you are not in a court. You are or will be in a mediated setting trying to reach agreement about what will work for your DD. Good luck 