I honestly do not know how you will prove best interest, given that you are not looking to move the children's school for almost two years.
1. The children are doing well in their current school.
2. Later start time is not going to be a factor. It may be a negative, in fact, if it interferes with Mom's ability to get them to school and then herself to work. If she requires before-care, you may find yourself on the hook for that.
3. Jr High is almost four years away. You have no way of knowing what will happen in the space of those four years, either in your life or in the school district. What they have now, they may not in the future. Many districts are cutting back on options as a part of necessary budget cuts - in four years, that Jr High may well be no different than the other.
ETA: Just a note about the provisions being easy to google. Perhaps they are. However, you need to remember that everyone here is a volunteer. Most with other jobs, other commitments. So we really don't want to go off googling information that a poster could have just as easily provided for us, regarding their situation. One reason why we often ask posters to keep all of their questions/comments to a single thread - then we don't have to go back and forth, looking for back story. THe easier you make it for us, the better we can help you. Make sense?
4. Why is making friends in your area more important than making friends in Mom's area? Enroll them in some sort of rec program in your town if you want them to meet more kids local to you. (Although, honestly, 7 miles isn't a hardship in terms of getting together with the kids they already know...)
5. Where your oldest goes to school is moot. She will be graduating HS (or close to it) by the time the twins go to Jr High - so they won't ever be in the same school anyway.
6. School statistics? Statistics can be made to show most anything one wants them to. One can also argue them in many ways. You tell the court that your school has better test scores than hers. She turns around and argues that your school spends more time teaching to the test instead of teaching how to learn, interpret information, think for oneself. You say that your district has more money to spend on schools. She argues that your district uses it for "fat" instead of education (hiring more administrators, instead of hiring more qualified teachers and buying books.) And so on and so forth.
I really don't see you getting far here w/o convincing Mom it's a good idea.