Do not feel bad. You are doing the right thing.
The landlord was like the neighbors are gonna be mad if it’s their cat. Well I don’t want my defenseless chickens attacked either. I guess some would do worse things to the cats.
Also, our no kill shelter... refuses to take any owner surrendered animals.. leaving town folk without many options if they get in a bind. What good is a no kill shelter if it doesn't do much to help the actual community?? 
Most people don't know the whole truth about "no kill" shelters.
I guess I don’t understand what you mean by dispatch? I could never harm an animal unless it was a threat to my pups or child. The shelter I take them to said they only kill for behavior issues or extremely poor health. Was that a lie? I didn’t think I was doing anything illegal. Really confused now I guess.Because no kill shelters are still allowed to kill up to 10% of their animals and life in a shelter isn't much of a life at all... it's more like dying slowly in a concrete prison.
That being said... You could protect your birds without trapping... E fence goes a long way towards that. Or even a secure shelter and covered run. I have just about every predator known to man... if I started trapping it would never end.
And also... I'm sorry that comes off rude!
I'm GLAD... that you are doing your best to protect your animals... in a way that you view as kind as possible. It's fantastic that you are not simply relocating to somewhere else that is not your problem, but doing the responsible, legal, and most appropriate thing that you are able.
While I might consider dispatching as the least cruel... we all have different thought processes and we ALL have to live with OURSELVES! All in all... even if I wasn't protecting chicks... If I had a stray cat problem... I would consider trapping and doing what you are... simply as a means of population control. Some of our local shelters trap, fix, and return them to live out their lives as they were but prevent future litters.
Hang in there. No need to feel bad. At least the cats should get a good diet and proper parasite control while in there. And if they are lucky maybe they will even find a pet home! I'm not in agreement with it... but our county shelter has even gone grain free for their dog food. Most places really are TRYING.
Don't trap if you aren't taking to a LEGAL drop off location or WILLING to dispatch. Relocation is often illegal and ALWAYS cruel.. dropping off in another animals territory with no known food, water, or shelter.
