I've read the various threads that pop up regarding private vs public and I find it laughable that many posters are so binary, all private schools are lumped into the same category or alternately all public schools are lumped in together. There are good schools and bad schools, private or public. It's a no-brainer. Only an elite few private schools are expensive ruggers and pony. Similarly not every public school is next door to a housing commission. These generalisations are ludicrous.
In response to the OP if you are employed in a non profit organisation you can salary sacrifice for nominated specific agreed costs, from memory, mortgage, school fees, super etc Thus making your school fees tax free.
If you are looking for tax free Kings fees I suggest your accountants would already be on top of it, but then you probably don't pay any tax already nor would you be on SB.