I agree with Katrina. Your daughter is not ready. It could be physical but it's most likely just developmental, although now it has become emotional. She's not ready. If she told you she wants to be a little girl, that's a big cue. If you're just surmising it, you may be right.
Look at all the things she has already done early. The potty is late on her list. Lots of other kids use the potty early but are late talkers or late walkers or late to learn the alphabet. I don't even like using the "late" and "early" labels because they imply a judgment, which I don't intend. She will catch up.
You cannot win this battle. And it is a battle now - a battle of wills, of power. Stop talking about it. Put her back in pull-ups and leave her alone. Put a beach towel or waterproof pad in the car and 2 plastic bags (one for poop and one for dirty clothes, and if you are out someplace, change her without comment (as you did when she was 1 and 2 and 3), and move on with your day. At home, change her. Say nothing - try to keep your dismay under wraps. You did that for a month - but you expected a change in her behavior at the end of the month. That was, apparently, an unrealistic expectation.
Your daughter is not alone. A lot of 4 year olds aren't there yet. I promise you that she will not go to first grade in diapers. If she cannot go to preschool because she isn't trained, then she doesn't go to preschool. It's not the end of the world. You go to the library and you go to museums, you read books and do crafts at home, you develop every other aspect of her intellect and creativity, and you do play dates with others to give her socialization.
You cannot have her pooping in the house any longer. And the constipation is tough - a lot of kids hold their poop because they don't want to use the potty, and it's not healthy.
Let it go. You have to take all the other aspects out of this - the power, the frustration, the discipline, the cleaning, etc. You cannot force this issue. She has full control - all children do - of potty, eating and sleeping. It has to stop being a battle ground. I know you are sick of this - I was too. My son was well past 4 when he was reliable during the day, and he was 12 before he was reliably dry at night. (Yes, we saw a pediatric urologist and actually had him on medication which, by 8, was a huge relief to us all so he could sleep at night.) So I hear you and I feel your pain. Please trust me - in your mind, pretend she is 2. She cannot change this, not now. You have to stop expecting it, and just wait for it.