EleanorRubysmum, I am sorry this must be hard for you.
In this case I would just go with your instincts. If you fear CM doesn´t like your DD, that should be a reason to change.
Change doesn´t have to be too stressful. It can be a huge relief for everyone involved.
Your DD sounds like a normal 3 year old. My DS NEVER wanted to put his shoes on, never wanted to wear his coat, was very defiant in general. It was part of "Terrible Twos" (which ends , sadly, roundabout 4 in my experience). It requires a bit of patience.
I think that as your DD has been with this CM since very little, she feels "free" to behave like that, normally this sort of behavious is saved for home. So a change might work, a new kind CM might mean your DD behaves a bit better (my DS was good with his CM but saved his defiance for home!).
(PS a friend of mine has recently been taken into hospital with puerperal psychosis, I am really worried about her. Is there a good website you can recommend for me to read up on this condition? It happened two weeks ago, and the meds haven´t started working yet. How long does it normally take before you can function again? Sorry if you don´t want to talk about this! Feel free to ignore. Have been looking for a thread on it here, but not found one)