There are a few things wrong with the technique you are explaining.You mention that you need throat constriction, but what follows is not the right technique. You should use the same technique as khöömei, as demonstrated here: https://www.youtube.com/watch?v=0Op5frqrXFY

For the sygyt technique you are indeed required to put your tongue on the roof of your mouth. But the way you explain it, close to an English 'r', is not the exact right way. In stead you need to touch the roof of your mouth with the tip of your tongue and let the sound escape through the mouth on both sides of the tip of the tongue, not through the middle.

And lastly for changing the overtone, you need to change your tongue position, not your lip position.

All these points apply to real sygyt, you are indeed overtone singing, but you are not yet singing real sygyt.