I for one have great sympathy for those who want to ensure their children complete the school term before moving (although that was NEVER possible for me or my wife’s families when we were growing up) and believe they should reach into their savings to finance “staying on” for the time required. I’m sure they have the assets (and, if it is really that important to them there are always loans).
The political appointees are in a completely different class.
Changing schools during the school year makes you stronger. I know from multiple experiences.
One year more or less makes no difference unless the parents create an everlasting stink.