Feral cats have few natural enemies. They need to be kept in check, if the cat is in my coop once he will come back. Regardless of the reason we have feral cats and I agree with you on that. I will do what I have to do protect my birds.
Killing a feral cat is a shame but letting it live is a bigger shame. It will destroy baby birds, aka kill them. It will kill chipmunks, squirrels and other small animals and surely will kill chicks. It knows where the food source is now. Besides leaving this cat alive allows it to produce more feral cats. A feral cat lives about 3 years ( not a like a well cared for house cat). During that time they will have 3-4 litters a year with 4-6 kittens a time.
By not killing that cat you are turning loose 35 more feral cats in the first year alone, the babies have babies at 7-8 months.
I understand your feelings for the cat, but reason, to me says get rid of the cat, the sooner the better.
I do not take pleasure in a killing like this, but I would do it. My opinion and I understand yours may be different.