Exercise in fasted state ?

I have had a pretty severely reduced appetite since September. So far, my doctors do not know what is causing it. Some days I have to force myself to eat more than 200 calories a day! It is my only symptom, there is no pain, and my blood work all came back healthy. I've lost a pretty considerable amount of weight (27 lbs between Oct. 1 and Jan. 15) which is fine since I'm overweight to start with and it would take a long, long time before I ever have to worry about being underweight.

What I'm concerned with is losing muscle mass and fitness because of this. Even though I've always been overweight, I've always been active and fit, and I enjoy exercise. I love to dance, and go to dance class workouts fairly regularly. Since this appetite/weight loss though, if I exercise for more than about 40 minutes, I feel terrible--exhausted, naseous, headache, and I'm completely mentally out of it. (Under 40 minutes is fine!)

Basically my question is what is the ideal way to approach physical activity when your body is in a severely fasted state? My goals are to maintain whatever muscle mass and cardio fitness I have if possible, and to enjoy movement. Am I doing myself more harm than good if I try to exercise? Are there any strategies I can use to make this work? (And please don't say eat more--I'm doing the best that I can!)