We had Mink in the canal alongside the chicken run. It only had a 3 foot high piece of 1" chicken mesh on it and we had no problems. I wonder if Mink have the same problems with chicken wire as Pine Martens? PM's can't fully retract their claws which then get caught on the wire I've read. Big problem in parts of France where they use the rubber parts of cars to floss their teeth -brake pipes are rubber! They put chicken wire on the floor and park their cars on it to stop the problem.
I would have thought Mink wouldn't go near your dogs Chickenfan.