All the trophies "do something with races" work with tournament races:

Connected 1
Finish 10 online races ==> online tournament races count

Gemini
Complete 20 two-player, split-screen races ==> tournament races count

Bling Brigade
Finish an eight player online race with all players using ships that are displaying Campaign Skins ==> online tournament races count

Motion Master
Win a Campaign or Racebox race with the motion sensor option set to 'pitch and steering' ==> tournament races count

etc

The tournament AI seems easier than Single Race AI. The whole point of this thread is: if I set some Phantom class Racebox tournaments will the races count for Arcade Perfect, like all other trophies work with tournaments?