Unfortunately the district boundaries do in fact determine the school your children will attend. I would contact the district directly to confirm the school boundaries for your new school and explain the situation. It is a difficult issue, but I am sure your children will be able thrive in either school setting. If you want to learn more about all the schools within your district, check out our SchoolFinder tool.
I'm not sure of the school district you're in, but contact the school and be forthright with them. These days, school choice is more than a buzz word and you may be able to use open enrollment to allow them to finish where they started.
The school has to have available seats, and every district handles the innerdistrict transfers differently.
But you should ask, since you have nothing to lose. Good luck, and let me know how it goes.