While I wouldn't discourage you from taking probiotics if you feel it helps you, I very much doubt candida is the root cause of exfoliative chelitis and that exfoliative chelitis is directly connected with intestinal issues. The alternative health world likes to attribute almost every illness out there to candida (I know because I have had Chronic-Fatigue-Syndrome for over ten years and been on multiple anti-candida diets on the advice of alternative health practitioners with no success whatsoever). The idea that multiple illnesses are caused by candida is not supported by scientific evidence.
I think we can agree, though, that EC is a localised inflammatory disorder of the lips and/or mouth which may be influenced by systemic factors. So anything which influences your general health is going to have an impact on EC. There are two scientifically plausible ways I can see candida could be connected to exfoliative chelitis (there might be other ways, but I just can't think of them).
1) Candida is an opportunistic infection connected with lack of skin integrity in the lip region.
2) People with EC have an autoimmune reaction to candida or, alternatively, a lack of immunity to candida, similar to that seen with sebhorric dermatitis and the fungus Malassezia. I don't think EC can be directly equivalent to sebhorric dermatitis, though, even if it has very similar symptoms because seb derm effects areas rich in sebum and the lips have no oil glands.
Also, if EC is the result of candida overgrowth, how come it only shows up on the lip skin and not other areas of the body? Thus, option 2) does not really make sense unless there is another factor causing lip injury. There is actually an illness called chronic mucocutaneous candidiasis which results from an immunodeficiency to candida. See this case report: http://www.vgrd.org/archive/cases/2007/cmc/cmc.htm
Notice how this little girl clearly has a candida infection of the lips. Yet she doesn't have EC.
So how can people with EC, with no documented immunodeficiency to candida, have EC as a result of candida infection? It just doesn't make sense from a scientific standpoint.
I think maybe your lips cleared up on the Water Fast because EC is connected to dry mouth syndrome and sipping water frequently might have alleviated this. This is just my hypothesis but thought I would mention it.