Originally posted by Nascarman:

I put my chickens in a raised up shed so that mobs wouldn’t even aggro on the chickens because they couldn’t reach them (well, the base is on an island so mobs can’t spawn anyway but you get the idea).

When it comes to a fire source, hanging braziers is the way to go. You can place them in the middle of the roof and get complete fire coverage easily and there’s 0 risk of the chickens dying to fire as the fire is floating above them.

a simple walled-in campfire works too.
the warmth still applies it aoe despite being walled into a chimney.
took me about 5 minutes to build a chicken coup that worked flawlessly, not sure whats causing OP's issue.