Yep would look at the motor mounts first.
Then the fairing brackets.
Just to be sure.
But yours does seem a little unusual, smooth at idle
and bad running at speed.
Nothing else remove the mirors and try stem mounts.
I have the stem mounted mirrors on my 06 EGC never had
a problem with the mirrors.
I've let my RPM's drop to 1800 many times, and gave it throttle and it picked
right up in any gear. But then it's tuned for best performance from 1800
to 3500. where I do 99% of my realistic riding.

 

You may want to have a look at the brackets inside the fairings. It is a week point in the 06SG. If one is broken you will get a lot of vibration