Responding to your neurologist's misguided view of things:To put it simply, she said that the mechanisms underlying chronic fatigue are probably comparable to the mechanisms underlying chronic pain (as in: pain that can not be explained by looking at the body part that's in pain).
Chronic pain is simply pain that goes on for a long time. Your neurologist is wrong to assume that all chronic pain can't be explained by looking at the body part that is in pain. Chronic back pain may in fact be due to deterioration in the spine, chronic knee pain may be due to mechanical issues in the knee.
They are also incredibly wrong to assume that all chronic fatigue is due to some sort of faulty central sensitisation. There is a whole range of reasons why someone might be chronically fatigued, many of them very treatable (and definitely not amenable to BPS treatments). If your neurologist doesn't understand that, I'd definitely be looking for a new neurologist.