There'a a quick underneath the nail, and if you cut this it will bleed for ages, and be very sore. I'm not sure how much a vet will charge, but I don't think it will be much.
I have just this week had the vet to cut the claws of my 21yr old cat, who no longer wears them down by scratching on a tree as she used to, so they need cutting every six months or so. The vet charges �4.50 for a cat, so it might be �6 or �7 for a dog, and well worth it. Also I think he would show you how to do it. Apart from anything else, the dog's comfort comes before Christmas. couldn't you treat it as his/her Christmas present.