My mum suffers from chronic pain and she exercises everyday. It's a massive battle, but it really improves her mood and is fundamental to maintaining her health because of the condition she has.

If it makes you feel better, even for a little bit, then do it, especially if resting makes no difference. Part of the reason I run is to keep the mental demons at bay and I find even a couple of miles helps.

Maybe you could combine exercise and rest with a gentle swim if a run is too much?

I know from my mum's experience with pain a positive attitude is so important to leading a fulfilling life. I imagine it's the same with fatigue, if running helps you feel positive keep at it!