I don't think it's a question of how safe the anaesthetic is, more a case of how hale and hearty the dog is.
10 years ago I was told by a vet that my 14 year old collie absolutely had to have a growth removed from his leg, even though it had been there for some time but was benign. Tragically, he passed away under the anaesthetic. He, although elderly, was otherwise quite well, but I believe that his little heart failed during the op. I still feel dreadfully guilty about having agreed to what the (very young) vet had advised us.
Sorry to sound like a killjoy, but the risks have to be faced and assessed. I sincerely hope that your little Westie comes through unscathed.