The DoE isn't likely to make an exception - if there isn't space for out-of-zone siblings, there isn't space. You will probably be told that if it is a hardship to do multiple drop-offs that you can move all kids to your zoned school (I don't think the DoE will care that you don't like your zoned school or about the disruption for your older children to move schools).
If you want to try to plead your case, I believe your district's enrollment office is the place to turn. That contact information is buried somewhere on
schools.nyc.gov.
Good luck.