You never set the PATH variable to include java anywhere. Your home system knows where to find a copy of java (because it's most likely installed on your machine too) but the school system doesn't.
So you can either try and change the path variables on the school computer (which you may or may not have the rights to do, and it leaves a trail they might not like), or you can make your bat file refer to the copy of Java on the flash drive by it's full path name, i.e. %CD%\mcp_data\Java\bin\java.exe
or whatever the path of the java executable is.
I recommend using the full path name since it's in a bat file, so the fact that it's longer doesn't really matter.
See below.
@echo off
Title DO NOT CLOSE THIS WINDOW!!! Closing this window will force close MineCraft
set APPDATA=%CD%/mcp_data
"%CD%\mcp_data\Java\bin\java.exe" -Xmx3072m -Dorg.lwjgl.opengl.Display.allowSoftwareOpenGL=true -jar "%CD%\mcp_data\launcher\MineCraft.jar"