Gordon, I think it is ambivalent. Neither interpretation is wrong but there is a preferred approach.
I think "delicate and refined distaste" refers first and foremost to the doctor's ATTITUDE and frame of mind, and this description is a development from the earlier "seemed disturbed by the war". The patient, or Hemingway, sees this with his eyes. He sees the face; he notes the physical body frame and the body language - "thin quiet little man" who "seemed disturbed".
The reader who can visualise (see it as a cinematic sequence of images) the scene understands it best. Hemingway has put in a number of adjectives to help the reader conjure up this cinematic scene:
thin
quiet
little man
disturbed
delicate
refined distaste
fragile delicacy
exhausted
One can see this doctor vividly. He gives the impression of having a delicate temperament. His demeanour is refined but disturbed. This is the first visual image Hemingway sees.
Then one sees him in action. He extracts some steel splinters. The principal thing that Hemingway notes is the distaste, somewhat concealed and yet perceptible; hence "refined".
By extension, from "quiet", "little", "delicate" and "refined", the reader would infer the style of his surgery, as a matter of transferred epithet, to be quiet, restrained and delicate. So Hemingway, and the reader, sees the face first, before he moves on to feel the surgery.
Note that the patient is under local anaethesia, and the afflicted part is "frozen". So the physical sensitivity is reduced. The scene is mainly felt visually. Hemingway sees the distaste. He sees the exhaustion. He sees the delicacy, refinement and fragility.
It is all about the doctor's frame of mind, not so much about his surgery or surgical skill. This is how Hemingway has painted, or filmed, the scene.
This is how the scene should be read.