There are e at least 3 cougars living in the largest town near me. They have never caused any trouble. Well, except some idiot sicc'ed his pit-bull on one and had to pay a lot of money for a lot of stitches for his dog. I don't count that as a cougar problem, though.
I've got at least one that lives less than a mile from my poultry and he hasn't caused any problems.
You might pen your birds up at night, which you should be doing, anyway. Then you make some noise if you are going to be walking around in the dark, to give the cougar some warning to move out of your way.