UPDATE:
Last night, we caught... the wrong thing!
So we set up a trail cam next to the trap (but we have yet to look over the footage) and we left one of our gates open, kind of to lure in the culprit. We did shut the actual chicken coop though, but the culprit(s) had access to the run.
Next to the coop door (which was closed), we set up the cage (last night) with fresh chicken thighs and some canned cat food. This morning we come and see a racoon! It was huge, and it started hissing and jumping around the cage right away.
I am positive that it was not the one that was eating the chicks because it would definetely not fit through the holes. But when we left the gate open last night, that's how it got in.
We were expecting a weasel or something small - and we get a giant, fat (probably pregnant) racoon!
First thing tomorrow morning we are driving it out of town.
We will set up the trap again tomorrow night.