Exhaustion and fatigue

I recently had my DLA to PIP assessment and have gone from high rate DLA mobility to just 4 points PIP for mobility component. I do have conditions which cause pain with mobility, but the main thing that limits my mobility is exhaustion and fatigue.

The assessor assessed my ability of how far I could walk on touching my knees whilst I was asked to raise them, whilst sat down and her own intuition. One leg was assessed as normal and the other as slightly weak. This seems to be a completely inaccurate way of judging how far someone can walk. I do have some "brute" strength in my arms and legs, but no stamina whatsoever.
I do have many medical conditions that cause fatigue and weakness (awaiting kidney transplant, vasculitis, Rheumatoid arthritis, heart failure, copd, to name but a few).

What I want to know is, is it worth me appealing on the basis of fatigue, exhaustion and weakness, or are these symptoms not classed as disabling (even though I know they are)?

I was given a glass of water during the assessment and the assessor could see that my hands were shaking and I struggled not to spill. All my points I scored on care component were related to my shaking hands. I was pleased that I scored 12 points for care, but very concerned that if I hadn't have been offered a glass of water, I may have ended up with no points. My claim form was filled in with plenty of details, examples and a fortnight diary. I don't believe any of the claim form was taken in to consideration, but the whole of my PIP was based on how I held a cup of water.