How to push back my school date before I start for AIT?

You can’t “delay” a school date. It’s not like a PCS where you can just show up a week or two later in the middle of the course and be ok. School seats are a limited resource. If you can’t make it to that school date, that slot is gone forever. You have to get a seat on the next class that has any remaining seats. That might not be 91F. That might not be till next year. Most of the school seats are already filled by Soldiers who are going to BCT so most of the training seats within 10 weeks have all been filled for a while. The ones within the 10 weeks of basic training are people who failed to ship or were dropped. Meaning you get what’s available not necessarily what you want.
Also, if you don’t swear in on the day you’re supposed you have failed to ship. That will have its own separate repercussions

It might be possible. You'll have to talk to your recruiter, and they'll have to send the request up their chain of command. After that, it'll all depend on your reasoning and available class dates, as CSM Darieus ZaGara said.

I am sure it depends on your reasoning and available class dates. You don’t want to wait yourself out of an available position. Recruitment never stops. If you took a job at any other company and said, oh yeah I can start in three months instead of tomorrow.