E.B.
This has nothing to do with how long your children have attended the school, or how you lost your house. It is simply a matter of school district requirements. Only students who reside within the school district boundaries may attend a certain school. Sometimes schools allow students from other districts to enroll, but that's if they have enough room, and if that kind of open enrollment or school choice exists as a policy in the town or district.
If it is essential, and a priority, that your son stay in the same school, you'll have to rent something quickly and produce a lease. It might not be a house that you like, but if the school is the number one factor in where you live, you may have to put up with an apartment or house that is not your preference.
Make sure to check your school district's boundaries on a map. A realtor can tell you that information, or the school website should have something about that. That way you won't rent a house or apartment, only to find out it's a couple of streets outside the school district boundaries.
Talking to the principal won't get your son any privileges. You either may enroll him in the district where you are currently residing, or rent/buy housing in the district where you want him to be enrolled. Of course your son can continue going to school; he'll simply have to change schools and attend the school that's in the district where you're currently staying. If you are staying with relatives and aren't paying rent or any utilities, you may have a difficult time enrolling your son in any school. Schools operate with tax money from legal residents. That's just how it is.