I hear your pooch is going to be fine, which is good news indeed. For others who may be reading this let me add that several years ago two of my own dogs were bitten by a single copperhead in our backyard. We took then to the vet ER in Matthews and they were treated successfully (though my wallet took a hit). We learned that copperhead bites are rarely fatal, even to dogs. Very small dogs bitten in the upper torso are at greatest risk but even they tend to survive. A good first step if your own dog gets bitten is to give him or her some Benadryl before heading off to the vet. An anti-allergy medicine like that is part of the routine protocol for copperhead bites. Otherwise, the vet is likely to give the animal an antibiotic and a pain killer. The swelling and pain usually subside fairly quickly. In our case I think the dogs were good as new in a couple of days.
By the way, young copperheads are likely to do more damage than older ones. That's because they have yet to learn not to use all of their venom in a single bite. Older copperheads have so when they bite they are like to pop their target with less venom (thus saving some for future prey). I'm not sure what to do with this semi-useless piece of trivia since I am not likely to ask an attacking copperhead for an ID card. But the point is to not think that a smaller, younger copperhead is less dangerous.
I now use a product called Snake Away around the perimeter of my yard to keep snakes out. You can buy it at Home Depot and Lowes. I'm not at all sure that it works but I use it anyway figuring that it can't hurt. It smells like mothballs. I haven't had any copperhead attacks since I started using that product. Draw your own conclusions.