I know what you're saying, but this is the way I look at it. Your pc prior to the r9 380 was quiet and at that time the only real significant heat source was the cpu (apu) and maybe a bit of heat coming off the vrms behind it. A cooler fan blowing directly on it and the front 200mm blowing through the case. Since you didn't change any settings in the bios when you added the r9 380, you added an additional significant heat source. The gpu does have its own cooler and vents somewhat to the rear but you have direct blowing fans on the gpu. Not a squirrel cage type fan blowing air front to back like the gtx 980ti has. Heat from the gpu will circulate around the case some, not to mention even though the cooler is on the far side of the gpu card away from the cpu - there's still heat radiating from the other side of that gpu right next to the cpu.
For a cramped itx case it looks to have decent airflow but I would still add rear fans to help 'guide' the air through the case past the cpu and other components. I'm not a big fan of stock coolers to begin with, my guess is that apu only it was borderline needing to kick the cpu fan up a notch. The additional gpu heat may have forced the apu fan to ramp up.
Here's a walkthrough of msi's click 4 bios, it might be worth playing with the fan settings to see if changing them has any effect on the cpu fan. It should. This would at least verify that speed control is working properly and that the fan isn't just stuck at 100% usage all the time.
https://www.youtube.com/watch?v=VTD_fBj12tA