Save the cat food for your cat.The mink is coming to kill,has it eaten any of the chickens? I have known a single mink, to kill dozens of chickens and ducks in one night,and never eat a bite.
Body grippers set in the trail to the coop should work---but--they must be placed in tight spots,where the approaching mink, cannot go around,over,or under the trap.
If the mink is entering through one or two holes--as was stated earlier, these can be set up with body grippers.If there are several entrance spots, block off all but one or two.
Let us know how you do.
Tom