Dog has severe separation anxiety. Best way to handle situation while owner is out of town?

I totally agree with Jessica about not haggling on prices. You are in NYC, and I expect that boarding is fairly expensive. Your house sitting rate seems incredibly low IMO.

Dogs with separation anxiety do much better in their own homes. The dog will probably go nuts if you board it and may even become destructive. Tell the owner you have done your homework and recommend staying there instead for the dog's well-being.

Note: Why is your holiday rate for house sitting less than for regular days? Holidays should be more.

I would stay firm to your prices for boarding (in your home, if that's the case). If you normally do home visits, then the client needs to understand that your rates for boarding in your home are a different cost. If the dog has severe separation anxiety, then I wouldn't be bringing a new dog into your home. I would ask about taking care of the dog in the owner's home. That way the surroundings are more familiar and the dog may be less likely to have anxiety.