Daughter wants to move in with her mom

From what you said in your post, it's obvious yours is the better environment for your daughter to be raised in. I had full custody of my two kids (7 and 2 at the time of my divorce) and my ex wasn't interested in having them at the time because she was too busy trying to re-capture what she thought she'd missed by getting married young (she was 19, I was 20 - we were married over 6 years before our oldest was born).
If I were in your shoes I would certainly be concerned about your daughter living in that situation, particularly because of the live-in boyfriend with kids spread out across the general population.
My first suggestion would be to consult a family-law lawyer to see what your options are to protect your daughter from your ex's lifestyle.
Another thing you might consider is letting your daughter spend the upcoming Christmas holiday break from school with her mom just to see what living there might be like. A couple of weeks might be all it takes. Compared to an occasional weekend, several weeks might be unbearable to either one.
Your daughter might also be feeling threatened by your upcoming marriage. Dads and daughters have a special bond. How has she bonded with your fiancé? Also, does your fiancé have any kids? What does the prospect of a blended family look like to your daughter? Kids are outstanding observers of behavior. While with your ex in the past, your daughter may have noticed how your ex's boyfriend's kids are sometimes treated better by your ex. It's possible she could be very afraid of losing your attention and part of your relationship with her after you get married. Counseling may help with this. Something to think about...