Kate, what about scattering powder or baking soda on the ground around the coop, it might let you pick up a print if it is coming around scoping things out at night. At least then you will know what you are delaing with.
Kate, just found this thread. I am so sorry for you and your birds.
I hope that you are able to get rid of whatever it is. My vote goes to Mountain Lion.
If I can help on the Spring rehatch, just let me know. Mi Pollo es su Pollo. ((hugs))