It seems the Embassy letter is no longer required by Immigration. The UK Embassy have just announced they are no longer issuing letters for new UK passports issued in BKK.
No, as the OP is home schooling. One issue with getting an Ed visa is an overstay in the kid's passport can cause a problem. Always best for children to be in Thailand legally.
Kids don't get fined or banned for overstaying but their passport is stamped 'overstay'. This shouldn't be a problem if they're not going to a proper school in Thailand. Personally I prefer my son to be here legally.