I just tested with Ifrit and got the exact same thing btw - 199 Burning Strikes but I think I lost one at the start as he ran into range after my Ruin. He also did exactly 200 autoattacks. I use Obey so no timing get whacked with OGCD usage. Also of note, if you change gear you need to resummon your pet.

As of now unless I see something pretty definitive otherwise - despite patch notes stating otherwise, spell speed doesn't impact the pets...