If he doesn't file within the 30 day objection period you will be granted an order to relocate. If he files after that, they may still make a court date and he'll have to show good cause as to why he didn't file within the 30 day period. If he has good cause they may allow his objection and...
Generally you have to have a change in circumstances in order to petition for custody change. The move or change in schools may or may not constitute. A lawyer who is familiar within your jurisdiction should be able to tell you if it'll be enough or not.
I understand completely how difficult...