Hi on the dual drive.
The sram dual drive is compatable with shimano brifters and any shimano sis changers of the same number. This is the advise from fisher uk who import sram.
You can discount the advise about shimano not compactable with an ihg as the dualdrive is set to shift at the same cable pull as a tripple shifter. Ie.

From my experience its usually down to friction in the long cable runs on folders and changing to teflon linned inners has usually worked to sort problems with the ihg shifting.

Else check the click box selector pin is not bent, loose.

The other time that can help is to increase the spring back of the cable with elastic bands,pen springs or a hair bobble.
This cable friction is a common problem with folders not just dd.

Hope my suggestions do the trick for you
I have teflon linned cables and telfon linned kevla cables on my 5 speed brommie and still need to use a hair bobble to get it to shift down well.