Here is the full list of all 13 Bedlam trophies - there are 10 bronze, 2 silver and 1 gold.
Complete The Game
Get a Score of at least 3500 in Rescue Run
Complete Rescue Run
Complete Planetfire Ascension
Wish Granted in Planetfire
Bring down the Super Saucer
Destroy the Armoured Car
Complete Rage-Quit Ractor without the Reactor exploding
Exit Star Fire
Kill 5 opponents in the Arena without dying
Defeat Lanzok and free the captive marines
Destroy the Gaian Comms Jammer
Completed the secondary objective