"Oh, she died and some lucky rat went to work "
You have a good reason to suspect a 'lucky rat' may have killed your chicken. Limiting access to the feed can make them attack each other or your chickens( the youngest & oldest are the most vulnerable)I wish you the best with your problem and sorry you're going thru this in the winter when its so cold.It really limits how much you can do
With respect, I'll argue with you on that one.
Chicks, yeah, even with plenty of feed available the rats will go after chicks. That is what started me on the road to creating our feeder which you can find the original origin thread here on BYC. But after installing the first plywood, inward swinging door feeder, protected by a 3/4" diameter round bar that slid up and down in a couple of electrical conduit clamps to keep the door closed, we didn't loose a single chick.
Adult chickens, no, rarely attacked by rats especially if they have proper roosts where the rats cannot get at their toes. If an adult hen dies or is close to death, then yes, starving rats might go after them if they are stationary.
The rats WILL attack each other, you were 100% correct on that. The first casualties are the young, cannibalized as soon the food supply is gone. Then the smaller ones until only the large rats that are more dangerous to try to take down are left.
So don't be worried about treadle feeders causing rats to attack chickens, were that the case there wouldn't be many treadle feeders on the market. Rats have four to five days to move or die assuming there is no other human supplied feed source nearby. In my case on day four they were staggering around in the yard desperately searching for food and my two dogs were having a blast killing them.