What's a precise translation of "they wouldn’t know if their backside was on fire"?
The government statement suggests PETA supporters only see things in their perspective, never care to consider other people's side, or look at the big picture, simply put - they are short-sighted (目光如豆), ignorant (愚昧無知) and don't know the suffering of the people (不知民間疾苦)

...wouldn’t know if their backside was on fire&quot;


...他們後庭着火(自己)也不會知道


(they don't observe and think about things around them)

後庭(backyard) is also slang for 屁股 (buttock)