well, first of all, you don't need a child psychologist...
she is 3, its the middle of the night, and its not the end of the world 
I know it feels awful, but its not really such a huge deal. She is right at the age, when she wants to do things herself, but sometimes can't. Even deciding where to go and wee can become a big thing FOR HER. But its your job to help her realise its not such a big thing.
however, at 1am, everything is going to get blown out of proportion, isn't it?
if the answer is for Daddy to do night time toilet visits, then that is what you will have to do.
by the way, when they are babies/very small, its the nature of the game that you run around after them. They can't do much themselves. She is growing up and exerting her will and her choice. It can be tough, but yes, compromise is important, as is setting reasonable boundries, so she feels safe.
you're doing great
go and give her a cuddle, and then get some more sleep