I am a major believer in the use of zip ties, especially extreme weather ones. However, I might opt for metal ones (if they exist?) or cutting small lengths of sturdy wire and twisting those through the hardware cloth and chain link.
Having said that, I am not the best role model. I have kept hardware cloth attached to a hog-panel run on one of my coops for several years using heavy duty zip ties.
HOWEVER, I am pretty certain that a determined mink or similar predator COULD chew through those if it so desired. The birds in that coop/run set-up are only out when I am home and are locked into the coop at night. The coop sits right outside my living room window, and I am pretty confident I would be notified of a predator's approach by my crazy terrier, Sir Henry the Loud (resident rooster in that coop) or the ever-on-the-lookout-for-everything Buff geese.
Good luck on buying all you need at the store on the first try -- an experience I've rarely had

.