I have the Barfly integrated mount and I am using it with a Fizik stem. I replaced my Cannondale C1 components with Fizik when I got a bike fit.
The measured opening on the Fizik stem through which the mount attaches is 6.5mm. This is enough space to hold the integrated mount in place. Not sure what the measurement on the stem you are trying to use are.

The Barfly page for the mount states it is compatible with the Fizik R1,R3 and 3T ARX II stems. Bar Fly Direct MINI stem mount ? Bar Fly

Fizik doesn't make 12 degree stems. They only make 7 and 20 degree stems: cyrano stem r1

Note: if you are wondering what the device under the direct mount is in the picture below is; this is the Di2 junction box. It makes the Di2 cabling a lot cleaner.