Hawk was my first turbine also. I like high side silhuette and rudder authority just like you. This has big influence on pure inputs it has. I think it is always a safe bet not to use HH on rudder. I think there must be something I have not yet found on HH mode just by looking how Havok Mephisto etc. new age sport planes. Both of my gyros are also evolved a step further now. Would be interresting to test some more but at this stage I still think gyro is not cure for aerodynamical issue. And HH has let me little disapointed so far. Hopefully I'm wrong it would be much faster to setup a good plane with computer instead of balancing and tweaking controls. One thing that is not always mentioned with gyros is servo travel speed their operate. It is feedback loop and speed must have a factor in it.

Your missing elevator authority combined with climb on high thrust suggest me your CG is more noseweighted than mine. I have almost opposite problem with elevator and I just reduced travel some more. I now have new and pristine plywood nose gear mounts and lighter tyre so balancing rearward than it used to. So my green one is now pretty close to my red that I liked a lot. Well weekend and RC meeting is coming.

Have fun experimenting
/Jyri