Indefinite contract with Private high school

I was wondering does anyone here have an indefinite contract with a private high school? If so how did you go around to getting it?

I’m currently working at a private high school under a direct contract with the school. I asked the English department head about an indefinite contract since I’m reaching the 5 year mark. But was told that the current school regulations does not allow foreign nationals to be full time teacher so the decision may be a matter that involves the board of directors.

I know that without a Japanese teaching license full time teacher is not possible. But I’m just wondering the chance of indefinite contract as an ALT at private high schools?

Any advice on how to get it is greatly appreciated.