If it's mostly the pick ups and drops off that are causing fear for yourself (I know there are other issues but in particular that one) then maybe you could always make sure you do them in a public place?
She is probably less likely to be aggressive towards you in a public place and if she was physical there would probably be witnesses that could provide evidence to the police. But personally I have found that most bullies (because this sounds like what she is) will only do it when they feel they can have the upper hand and at least being in a public place might make her wary of getting too mouthy with you.
Although honestly I have to say with regards to what you posted above:
"She just picked her up. Both of them. I was nice and said "She is being quite grouchy" she said "Oh, well she is not now".
If I am honest, I think you might be being a little over sensitive. While it might be an abrupt reply and said just to wind you up, I don't think it's anything to get too upset over. I'm sure it's upsetting because it sounds like another thing in a long line of abuse you are getting, but just try and ignore it. She is being childish, so treat her like a child. Smile nicely and ignore her tantrum just like you would a child.
Regarding the contact centre, if you don't have any worries about your child when she is with them, then I think supervised visits is going a bit over the top. There is no way she should be treating you like she is and calling you names, but if she is a good parent towards your daughter then supervised visits sound a bit extreme.
Your daughter lives with you and I'm sure as she grows up she will realise that any nasty things your ex may say about you are not true. Personally I feel pushing for supervised visits (especially if there is no risk to your daughters wellbeing while with them) is the sort of thing that could create longer term problems in the future.
Obviously this is just my opinion based on what you have said and I don't know all the facts. I really hope you can work out a way around all these problems. x