The court will consider the best interest of the child. Since this is a true joint custody case, any material change in either party's circumstances is enough to permit the court to decide custody. However, it will pretty much approach it as though deciding custody for the first time. From the facts you present, this is a case that could go either way. Sometimes, a judge will listen to the wishes of the child. All other things being equal, that may be enough to tip it in your favor. However, being as your daughter is only 11 years old, the judge may not give her wishes any significant consideration. If your ex doesn't have any children, then the court may find it best that the daughter live with you so that she can be close to her siblings. You may have to be prepared to show that switching schools will not be harmful to your daughter. With only 15 miles between you, you may be able to work out a more generous visitation schedule, which may help justify your ex being able to pay less than the charts require.
You have some really good family law lawyers available in your area, so be sure you get a good one. A custody battle is always difficult because there is no way to predict how a judge will rule if both parents are fit parents. As Mr. Eastman said in his answer, the best interests of the child will rule. On the one hand, a judge is not going to want to move her from her school. On the other hand, if there is a lot of family very close in the other town, that would be good for the child. If the child has a close relationship with her half-siblings (your other children), that will help as well, because courts don't like to separate siblings, even half-siblings. You also have the advantage of being able to be there for her when she gets out of school in the afternoon and being able to be with her all day in the summer and Christmas holidays, since you're a stay-at-home mom. That's always better than day care (of course, he might have someone like a grandma who could watch her while he's at work). All of these things are important to consider so choose your attorney carefully.
One thing you don't address is whether YOU want child support, since you said he doesn't want to pay it. While obviously if you have custody, he would have to support the child, sometimes you can work out something reasonable, such as him paying for her school supplies, school clothes, extra-curricular activities, etc. Some dads don't object to child support itself as much as they object to writing a check to the ex that the ex could then spend on anything. So if he's paying for specific things, he might not fight it as much. When I used to do family law, I could often get the dad to agree to help pay expenses and he ended up paying as much as he would for child support, but just in a more structured way. And once you get custody, if the need for regular child support comes up in the future, you can always go back to court to try to modify the child support.
Good luck with the move and the custody issues.
No attorney-client relationship is established with this answer. It is not to be considered legal advice, but is merely given to point you in the right direction and give you a general answer as to the law regarding the question you have asked.