I have a character who is human but doesn't generate their own body heat, like a cold-blooded animal. Is it unfeasible for a human brain to be operating despite this?
Yes. Biologicaly human brain is not that far from fish brain (birds brain is more evolutionaly "advanced"). So If you put it out of human body and provide it with right chemicals - it still will function down to freezing point (down to 4 degrees at least), but at very low rates of cognition. And will greatly heat up in process. You would need constatly remove heat to keep temperature low.
On the other hand human body "swithces off" brain at about 25 degrees (as it is noted in other answer) to save energy (brain is the most energy consuming single organ for humans)
32-35°C is considered "Mild Hypothermia", described as "Awake and shivering".
Symptoms of mild hypothermia may be vague, with sympathetic nervous system excitation (shivering, high blood pressure, fast heart rate, fast respiratory rate, and contraction of blood vessels). These are all physiological responses to preserve heat. Increased urine production due to cold, mental confusion, and liver dysfunction may also be present. Hyperglycemia may be present, as glucose consumption by cells and insulin secretion both decrease, and tissue sensitivity to insulin may be blunted. Sympathetic activation also releases glucose from the liver. In many cases, however, especially in people with alcoholic intoxication, hypoglycemia appears to be a more common cause. Hypoglycemia is also found in many people with hypothermia, as hypothermia may be a result of hypoglycemia.
36°C exactly isn't even Hypothermia. 32-35°C is sub-optimal, but your brain still works. 28-32°C and you’re drowsy and stop shivering. Below that the brain stops functioning.