Hi Mark,
I recommend Paraguay, where you could get a Permanent Residency quite cheaply (about 2000$ per adult if you use an agency) without the need to stay there permanently in the beginning to get the Residency.
Once you have the Residency, it is also easier to stay in the other Mercosur countries. I have not looked into the homeschooling situation there, but I guess the authorities don´t care so much about Gringo children and as long as you are willing to travel anyway through Southamerica, going to another country every three months or so, you should be fine anyway.
In general there is a lot of anarchist stuff going on in those countries south of the US, I think you will find your place there.
Personally, I prefer Asia, but the whole visa and property rights situation there for foreigners is much less favorable for long term stay.
Right now, I am still lazy and in hibernation, but maybe later this year I go to Paraguay and check it out.