Speaking as a former FTCer for two years (one where I had to start my own team, finding a whopping
1 teammate), I have a bit of experience for this one!
One thing I would suggest is the private school kid organizing a spot at a club fair, or making it a school sanctioned club. Play the animation for this year at an assembly, or just go around to science classes and start talking. You're bound to grab a few people. Plus now you have "veterans" to teach the "rookies" programming and building.
Associating with an admittedly far away FRC team can't hurt. Tag along with them when they do demos, doing some recruiting while you're helping them spread the word about science and technology is a great idea.
Getting other home school kids is a good idea, as it gives them more opportunities to do extracurriculars and interact socially than they otherwise would have. (mentioning this can't hurt your Inspire chances

) I'm not sure how to best do that, other than running newspaper ads or contacting various media sources (FRC teams seem to be good at that).
Good luck.