Mink are the worst, in my opinion. I had one last year for the first time. Live trap was a no-go. Leg trap with chicken livers, yep also a no-go. They *really* want a live victim, and they're awfully smart. Mine learned my daily routine and would show up after I left for work (when I do chores) which was about 6:30am and broad daylight.
If you really want to trap them what I found out was that you want basically a 110 conibear and a box to mount it in with a live prey cage on the back. Surrounding a caged bird with leg traps could work too I suppose, but I feel like you'd have to cover the traps with leaf litter or something.
I did order a tube trap online meant for gray squirrels and mink, but did not end up catching it, it had moved on at that point. I'm not sure how large of a loop they typically make, but once I reinforced my barn door (I had thought it was sufficient, but there were a few very small holes that apparently mink can use) and kept my outdoor birds locked in for a couple days, my mink moved on and hasn't returned in.......at least 6 months? Maybe more. It was sighted across town about a week after it stopped being able to access my birds.
With mink, about all you can really do is make sure that your birds are locked up like Ft. Knox. Which is hard if you don't have the time, money, and/or expertise/ability to build a permanent hardware cloth run with buried wire or apron-ed wire around the perimeter. Personally, I don't want a permanent run like that either, so my options are hope the mink doesn't return or stop having any birds outside period.
