I know exactly why your daughter does not want to wear a bra. At this age, she does not want to be the only one wearing one. It is hard enough for girls in older grades to transition into bras. She could also be a little uncomfortable about her changing body. Again, if she is the first one to start developing, it is scary and embarrassing.. Then if she is very modest, it can be hard for her to speak about it, even with her mom about such things.
Maybe you could get her a book that talks about her changing body and you 2 can read it together. Also tell her that her nipples will become more and more sensitive and it is going to hurt when she least expects it from the rubbing of the fabric.
We have a very good friend who has a daughter that experienced exactly this same early puberty. He daughter was mortified and would cover her ears, when her mother would try to speak with her. Finally, the mom told the rest of the family (in the house) not to mention anything about this to that daughter. They were act as though there were no changes. This helped the mom reassure her daughter, she was still the baby in the family and she was safe at home from any teasing.
She was also very athletic so they found some soft tank tops for her to wear. The mom also began shaving her daughters underarms and taught her daughter how to use deodorant.
One of the things the little girl was worried about was, what would she do at sleepovers? The mom told her to just keep the tank top on and sleep in it like a pajama top, or put a pajama top over the tank. She was able to find photos in a Seventeen magazine of girls wearing this type of thing for pj's.
This is becoming more and more common. By 4th grade, you will notice most girls wearing little bras or undershirts.