Here is the full list of all 12 Battalion Commander trophies - there are 7 bronze, 4 silver and 1 gold.
Complete all missions, buy all upgrades
Destroy 50 hostile vehicles using a car
Destroy the boss without stepping into napalm with squad leader
Heal 50 allies, who have red health indicator
Travel 2000 meters in Endless mode, in one go
Complete half of the way, using vehicle
Travel 1500 meters in Endless mode without using ability
Set free 500 hostages
Having collected all dropped coins in the field, destroy the submarine
Destroy 100 hostile jeeps
Destroy 200 hostile mortars
Destroy 1000 crates