A small body trap (conibear) will kill your dogs if they get caught in it. But, they are very effective on minks. Can't you keep the dogs away from your traps for just a couple of nights? Do your dogs have access to the "roost" area? Set your traps in the roost or run.
There are "offset" foot-hold traps that will probably not cause much harm to a small dog.
If you know the path he is taking, a snare will catch him. Especially a spring snare. They catch dogs also.
Another option is shooting the mink.
I don't know if a mink is big enough to set-off a motion detector. I used one of those motion detector security outdoor lights. I screwed in this receptacle thing, that has light-bulb threads, with a plug-in receptacle. Then, I ran an extension cord from that receptacle, into the house with a buzzer plugged in. When the coon (in this case) set off the motion sensor, the buzzer in the house went off. I went outside and shot the coon. Worked like charm.
Wouldn't bet any of my chickens on a radio keeping a determined mink away. The mink already has had several meals there and will return.