I don't really know the precise legalities of this, but am sure that no-one is going to be checking up on you that precisely.
Especially if you write to the school now, announcing your intentions and explaining that you expect to leave around (early/mid/late likely month) and given the needs of final departure family administration you are therefore planning your DC's last day in school to be (date). That should give you enough wiggle room especially as you say you anticipate travelling a bit in the run up to final embarkation.
It would, I think be rather tactless if the DC were hanging round at home, rather than being in school, as their mates might spot them and bubble you. But you'd be gone before anyone had a chance to query what you were up to officially.
Depends on the age of yr dcs but ime, kids are best in school otherwise they tend to get bored and under your feet, especially when you have so much to do when moving country. I would think that, as the USA schools are normally off from end of May, you could have a case.... Enjoy the 12 weeks of summer holidays!
We did this years ago - no problem at all. We were very fortunate in that the UK school held the place open until the end of the academic year(we went in March) in case things didn't work out as planned. In the end we stayed 8 years.