A reader asked this question, and a summary of what we find in the literature is likely good. Whether it applies to you depends on what bacteria genus you are high in.
What is a [proper] ketogenic diet – The ketogenic diet is a high-fat, adequate-protein, low-carbohydrate diet [Wikipedia]. It was first used in the 1920s, “outside of paediatric epilepsy, use of the ketogenic diet remains at the research stage“.
- “A recent study by our group has demonstrated that a brief ketogenic period, if followed by a longer period of correct Mediterranean diet could avoid this yo-yo effect” (Paoli et al., 2013)
- “This rapid and marked expansion in the representation of the Bacteroidetes, achieved after a 24-h fast,” 
- “fasting was associated with a significant increase in the proportional representation of the Bacteroidetes [from 20.6% (fed) to 42.3%”
- “A ketogenic diet did not produce a notable change in the ratio of Bacteroidetes to Firmicutes in the cecal microbiota compared with CARB-fed, untrained (or trained) controls (Fig. S2C). However, unlike fasting, a ketogenic diet was associated with a significant decrease in bacterial diversity (P < 0.001; Fig. S2A). “
- “KD reversed the elevated Akkermansia muciniphila content in the cecal and fecal matter of BTBR animals.”  – only a subset of CFS patients have high Akkermansia, most are low. 
- According to the above chart — Bifidobacteria is lowest on K.D.
- Lactobacillus and Bifidobacteria decreases over time on a low-carbohydrate high fat diet. 
- Lactobacillus and Bifidobacteria increase over time with high carbohydrate, low fat diet.
- “In contrast, the Roseburia spp. and Eubacterium rectalesubgroup of cluster XIVa and bifidobacteria decreased as carbohydrate intake decreased. ” 
With repeated studies finding a decrease of bifidobacteria and lactobacillus with a Ketogenic diet, I do not believe it has any clear benefit and may encourage further negative shifts with some CFS/IBS/FM patients.
CFS Patients appear to universally have very low or zero bifidobacteria, lactobacillus.