Just to explain the Kirayt Joel
There are laws in every state who can form a school district and there are laws in NY. You cant decide that your house is an independent school district and therefore not pay taxes.
Kiryat Joel did not meet the minimum requirements for an independent school district in NY so they wrote a law abut school district creation that only legalized this district in Kiryat Joel and nowhere else in NY (A seperate school district for obviously chassidim based on religion) that was found to be unconstitutional as the only place in NY it applied to was Kiryat Joel so they re-wrote the law and it still only applied to Kiryat Joel and that was declared to be unconstitutional as well. Finally they re–wrote the law a 3rd time that applied to about 5 places in NY (The others did not seek an independent school district) and that was found to be constitutional.
I dont know the laws of school districts and certainly not in NJ, but its unlikely Lakewood could be split up into 2 school districts especially since everyone lives together (unlike Kiryat Joel where very few if any minorities live)