I use 1/16 7x7 cable, (thinner cable may cut skin), loaded heavy, must have an inline swivel as mink are twisters and rollers not strong pullers. You want to neck snare the mink, then no more fur damage than on a coyote or fox. I have tanned many snared mink and never had a problem. I have snared and sold several thousand mink and have been docked, price wise, on probably less than 1%. The key is to have a FAST snare so as to not let the mink walk through and hip catch it. Notice the lock position in the picture, I want that snare to drop at the slightest touch. I usta' tell guys at conventions that by the time a mink pushed hard enough to fire your conibear with a loop trigger wire, I already had him dancin' in my snare.
About 20 years ago I had snares set in New Jersey, within 20 yards of the New York state line (No snaring in NY). Just on the NY side of the line another trapper had several conibears in the same high bank trail. Over several weeks I had caught about 20 mink. I met the other trapper there one day and he was dumbfounded why the mink would avoid his conibears and go into Jersey and get caught in the first snare. He said he had caught 3. I went over with him and looked at his sets and they looked good, but the tracks in the snow showed that mink would go over or around them no matter how much he tried to block them down. It was snow conditions so not possible to blend in the conibears with grass, he did the best he could with sticks but it was not enough.