Rear dew claws rarley get in the way and cause a problem. Sometimes front ones do. It's not even required to remove then per the breed standard. If the vet told you HAVE to have them removed I would think your vet is looking for a new Mercedes this year. Save the money and put in a real emergency fund.
I own a Chow also, and I would not recommend having the dew claws removed. Just keep them clipped like you would the front dew claws. Why did the vet say to remove them? My Chows have not had back dew claws, but when I asked about removing the front ones, it was a strong recommendation not too, it's costly and very painful for your dog, and no real reason to do it, other than we don't have to mess with trimming that nail. It is a difficult nail to trim.
A dewclaw is a vestigial digit of the foot of many mammals, birds, and reptiles (including some extinct orders, like certain theropods). Dogs almost always have dewclaws on the inside of the front legs and occasionally also on the hind legs. Occasionally some dogs will even have more than one dewclaw on the same foot; often at least one of these dewclaws will be poorly connected to the leg and in this case it is often surgically removed. There is some debate about whether a dewclaw helps dogs to gain traction when they run because, in some dogs, the dewclaw makes contact when they are running and the nail on the dewclaw often wears down in the same way that the nails on their other toes do, from contact with the ground. The dewclaws are not dead appendages. However, in some dogs these claws may not appear to be connected to the leg at all except by a flap of skin; in such dogs the claws do not have a use for gripping as the claw can easily fold or turn. The argument for removal states that dewclaws are a weak digit, barely attached to the leg, so that they can rip partway off or easily catch on something and break, which can be extremely painful and prone to infection.
If you are having the dog fixed at the same time then they can remove them at a low cost. Otherwise he is only recommending it, it doesn't mean you have to. I have seen them getting stuck or pulled off by accident, but that is rare.
When I look up into the sky, I think to myself, Wheres the ceiling?
The only chronic problem with dew claws that is definitely going to happen (in a breed where they don't touch the ground) is that the nail is going to grow too fast. This causes it to either curl back and dig into the dog causing a painful sore or it will grow just long enough to break causing a very painful broken claw. There is no reason to remove the hind dewclaws unless you are unable to trim the nails as often as they need it. As already stated it is not common for the claw to catch on things and tear. You would be inflicting an unneccessary surgery on the dog.