You do NOT need to kill the hardsuit proper, only the person inside. The flamethrower can be found at any Supply Depot on the map (locate these with R1, they're the white circles) for 350 CP and can even be picked up off the ground, provided a player (including you) died with one out.
Again, once you get credit for killing the player, you've gotten credit for one kill towards this trophy. The suit does NOT need to be destroyed.
Mine didn't pop until I backed all the way out of the game lobby, so don't panic if you keep count and it doesn't appear during the match.