The relentless itch that escalates when the dog is stimulated suggests Heart Fire is driving it. Qing Ying Tang is a formula that addresses Heart Fire and fits that full pulse.
So, too, do Shao Yang formulas. You’ve seemingly tried them, but I find they don’t work that well in skin issues unless paired with a Blood tonifying and moving formula like Si Wu Tang or Yi Guan Jian.
Lastly, if this dog is still on processed food, they may have Damp Heat, whose pulse can occasionally come across as Full. If a move to real food aids this dog, then Si Miao San, Long Dan Xie Gan Tang and Long Dan Er Miao San are all worth a look.
Hopefully, this helps you out and the dog gets better soon!