Shifting Youtube livestream from public to private mid-streaming?

I am planning on streaming to Youtube and embedding the stream on my website. Users on my website are paying members.

I would like the stream to be public and free on Youtube for the first half hour, though. But after that duration, I would like to be able to say goodbye to the free-watchers and shift from public to private setting on the Youtube live stream. Then I can continue the stream and it is only shown on my website and not accessible on Youtube anymore.

Is this possible without ending the stream?

An alternative to switching from public to private while streaming is to stream to my website in another way outside of Youtube. Then I can easily manually end the stream on Youtube when I want to and keep only the other stream active.