Before you make the announcement - Surprise! We're home schooling Lavinia! - just move house. Move house to a town, state or country where nobody knows you. Yes, it will be a lot of work, selling the house, buying a house, resigning from your job, packing all those boxes. And as a Slacker Homeschooling Mum, I don't normally recommend a lot of work.
Trust me on this one though. It will be less work than explaining to the entire world and her nosy neighbour why you have decided to home school Lavinia.
Even if you do manage the move and buy Lavinia a school uniform just to wear when she's Skyping with Nana, it's an unfortunate fact of life that you will probably end up meeting other people, whether you want to or not. Obviously, you can avoid actual friendships. I'm putting my faith in your inability to return calls/texts/emails or accept invitations. If Lavinia ventures outside for a moment though, you're done for.
"No school today, dear ?"
Teach Lavinia not to engage. Get her to practice an unnerving stare and the simple reply "No" delivered deadpan and followed by silence. It will save you from having to do it.
Once she's got that one down pat, teach her the next Slacker Essential Reply. If it somehow slips out that she's home schooled, after a prolonged interrogation by Shirley next door, there's going to be a follow up question.
"Is your mummy a teacher ?"
The correct answer to this is YES! Yes, my mummy is a teacher. Even if mummy is actually not. If Lavinia is an especially imaginative girl, she can elaborate on all the teaching awards you've received in your long career, but I don't advise it. Call her indoors and let her loose on the craft box instead.
Trust in Shirley to do her work and let it be known far and wide that yes, the new family are a little odd but at least the mother's a teacher! They'll still shake their heads - that poor child - but they are now unlikely to actually report you to a government agency.
You are now ready to begin home schooling.