Between doing other things today I took a moment and tried unscrewing the horn button again. It appear to loosen up a bit.

Enough so I was able to then slip a thin blade screwdriver under the horn button and see the thin lock ring on the brass cap under the button. With a machinist pick I removed the ring and got the horn button out.

Though the metal piece directly beneath the horn button will not separate from the button.

That is far as I could go because I had to go back to doing other things.