I agree with everything Tangle says.
You might find it easier to HE and keep them pottering along with whatever scheme of work they would have been doing in the UK, instead of putting them into a local school. (Perhaps their UK school would lend you books to take with you?) If they went into a local school in the US, they'd need to adjust to a different way of working while there, then play catch-up when they return because they will have "fallen behind" in whatever subjects aren't being done in the US - or aren't being done in the same way.
Plus there is the freedom you would all have if they weren't at school. You could make the most of your adventure abroad by going on road trips and visiting museums, historic sites and other places of interest.
With respect to local homeschooling laws, I'm sure you'll find an email group for homeschoolers in the state where you are going. The parents there may be able to point you in the right direction, and also may welcome you to their social activities as well as giving you tips on local places to visit. Ask whether the laws apply to short-term visitors such as yourselves: you might not count as residents if you are only there briefly.
Have you considered what you'll do upon your return to the UK? Talk to your children's school or to the LA. I don't think they are allowed to hold the children's places waiting for an entire term. So if the year group is full and someone else turns up who wants a place in that year, your child would lose the place. You might end up wanting to home educate any child who is without a school place for a while after you get back, in hopes that a place there will become free again. (The LA would certainly offer you a school place for each child upon request, but not necessarily at your preferred school, and not necessarily all at the same school.)
If the youngest is due to start Reception next year, you can secure her place by applying now, accepting the place you are offered, and requesting a deferred start for her. From next autumn all parents have the right to do that.
Have fun!!