Would a neurologist and cardiologist be able to diagnose an autonomic dysfunction or is it a difficult task? What symptoms would they look for?

More a neuro thing. unless you have to exclude intrinsic heart disease. Autonomic dysfunction presents in many ways depending on which parts of the autonomic nervous system are affected. One typically looks at symptoms and goes backward from there (in neurology the key question is where is the lesion followed by what is the lesion). My typical lecture on the topic is split into two parts as the topic is vast.