My interpretation of the last sentence is that if primary custody is shared between multiple districts (main example: two parents who are separated but share custody could choose the district between the two parents). Care of a guardian here doesn't mean helping with child rearing but the state assigning guardianship (so a sister, aunt, grandmother who is a full time appointed liasion).

You are certainly more than welcome to contact the district that you are hoping to go to and they can tell you whether they are able to enroll. I would strongly caution you against doing this under the table as it could be very disruptive for your kids to have to abruptly unenroll and difficult to explain. It's likely it will come up when your kids get to know friends.

I'm going to assume they are elementary from the school closing comment, but I would say your best course or action would be to complain to the district about this disruption in school closure to see if they can get you into your top choice (this is often Vartan if you need a suggestion). And while you're at it please remind the district how poor of an idea that is. I am sorry that your kids school is closing.