People like Dr. Mercola and Weston Price advocates the use of fermented sauerkraut for good reason — it is very high in lactobacillus bacteria. Lactobacillus bacteria is generally good because it kills off many other bacteria and helps a normal microbiome to be stable.
The problem in CFS is that CFSers are very low in E.Coli   – a species that Lactobacillus tend to kill off.
First, what species are in Sauerkraut?
The species in home made sauerkraut changes over time, DNA Fingerprinting of Lactic Acid Bacteria in Sauerkraut Fermentations including
- L. mesenteroides
- Weissella sp.
- L. citreum
- L. curvatus – inhibits E.Coli 
- L. fallax
- L. plantarum — inhibits E.Coli 
- L. brevis
- L. argentinum
This article Antagonistic effect of Lactobacillus strains against Escherichia coli and Listeria monocytogenes in milk., describes the general issue at play.
Increased D-Lactic Acid Intestinal Bacteria in Patients with Chronic Fatigue Syndrome  “This study suggests a probable link between intestinal colonization of Gram positive facultative anaerobic D-lactic acid bacteria and symptom expressions in a subgroup of patients with CFS. Given the fact that this might explain not only neurocognitive dysfunction in CFS patients but also mitochondrial dysfunction, these findings may have important clinical implications.”
Bringing up the E.Coli population is likely a significant factor for recovery, not reducing it.
“Patients with chronic fatigue syndrome (CFS) are affected by symptoms of cognitive dysfunction and neurological impairment, the cause of which has yet to be elucidated. However, these symptoms are strikingly similar to those of patients presented with D-lactic acidosis... this might explain not only neurocognitive dysfunction in CFS patients but also mitochondrial dysfunction, these findings may have important clinical implications.” [Full Text]. “Probiotics Provoked D-lactic Acidosis in Short Bowel Syndrome: Case Report and Literature Review” 
E.Coli produces a lot less lactic acid than lactobacillus, thus when they are diminished, the bacteria replacing them increases the production of D-lactic.
There is literature speculating that lactobacillus may help CFS (Medical Hypothesis 2003), as always, we prefer actual studies instead of “bright ideas with horrible consequences to CFS patients”.
I found some reports of treatment, and this one stands out – multiple antibiotics and probiotics afterwards (note the dosage of probiotics in 3 grams/day — far higher than the typical 4 mg in a commercial probiotic capsule — 1000 times more!!!!).
” The patient received kanamycin (Kanamycin Capsules, Meiji Seika Pharma, Tokyo, Japan) 1000 mg/d. … metronidazole (Flagyl, Shionogi & Co, Ltd, Osaka, Japan) 500 mg/d and kanamycin 2000 mg/d were administered for 5 days under fasting conditions. Polymyxin B (Polymyxin B Sulfate, Pfizer Japan Inc, Tokyo, Japan) 500 3 103 U/ d and vancomycin (Vancomycin Hydrochloride Powder, Lilly, Kobe, Japan) 1000 mg/d were administered over the subsequent 5 days. After the use of antibiotics, a purgative (Niflec, Ajinomoto Pharmaceuticals Co, Ltd, Tokyo, Japan) was used…..Overgrowth suppression was approached by starting synbiotics, specifically B breve Yakult (prepared by Yakult Co, Ltd, Tokyo, Japan) 3.0 g/d and L casei Shirota (Biolactis Powder, Yakult Co, Ltd, Tokyo, Japan) 3.0 g/d as probiotics, and galactooligosaccharide 8.4 g/d as a prebiotic.” 
Exercise and Lactic Acid
Exercise produces lactic acid which further compounds the issue and result in fatigue. See “Lactic Acidosis and Exercise: What You Need to Know” on WebMd. There appears to be no conventional treatment for Lactic Acidosis.
Root cause: Low Veillonella?
See this post: on what this bacteria does with lactic acid.
Alcohol and CFS – The E.Coli Response
While researching the above brief notes, it caught my eye that alcohol kills E.Coli, which would further swing a CFS patient towards D-lactic acidosis.