My big Lab was bitten by a venomous snake (it got away) - probably not a copperhead, probably a moccasin but could possibly have a been a rattler. The vet was very blase', said he'd never lost a dog to snakebite. It was very swollen for 24 hours then went away with no ill effects.
Copperheads don't have big fangs, they have to gnaw on you awhile to get any venom into you at all. My son was hit by a copperhead while running cross country -- it was apparently a "dry bite" because he never even swelled up.
Try benedryl next time your dog is bitten (unless a rattler, in which case I would get them to a vet asap). I used it and it took the swelling right down. Then antibiotics in case of infection. My old girl was bitten by something while my son was dog sitting while we were out of town. He didn’t notice anything, but when we got back she had a hugely swollen leg, so off to the vet. She drained a couple of cups of icky brownish fluid (probably blood) from the leg and then found the fang marks.
The old girl did ok, but never was quite the same. She died about a year later at almost 13.
susie