You are in luck! REM sleep is the sleep subject I have the most expertise in! This is because I have a disease that revolves around disordered REM sleep (narcolepsy with cataplexy).
So, first off, you may find it reassuring to know that REM sleep, contrary to popular misconception, is not "deep sleep." Deep sleep is actually NREM stages III and IV. These are the stages of sleep that are physically restorative to the body, and it is deficiency of these stages (due to excessive REM sleep) that causes people with narcolepsy to never feel rested, no matter how much sleep we get. However, and this is the caveat that I think is most significant in your case, REM sleep does have an important function. During REM sleep, we dream, and during this time we are "consolidating" memories and things we learned during the day. This is *possibly* one contributing source to the brain fog you're experiencing with your chronic fatigue syndrome, though it's important to note that brain fog can be a direct symptom of CFS itself, so it would be hard to separate these variables.
I would also be remiss if I didn't mention that research into sleep architecture is still ongoing, so there are many things about sleep we don't know yet.
I do think it may be possible for you to improve your REM sleep. I have another condition, generalized dysautonomia, which has some overlap with CFS so I understand what you're up against here. One thing that has helped me was getting one of those "fitness tracker" wearables that tracks your sleep for you by measuring small movements of your wrist. I have the Jawbone Up24, but there are many different options that can do this for you, in varying price ranges. Unfortunately, health insurance does not cover these at this time. What these devices can do is help you identify which nights you get more REM and which you get less, and record what time you went to bed and woke up; what you ate for dinner the night before; whether you had any caffeine or alcohol the day before; how much exercise you got and what time of day you did it; any variables that you think could possibly affecting your sleep. This kind of detailed tracking can help you fine-tune your sleep habits. You can log this data right here on MedHelp from either your computer or your iPhone using this tracker:
http://www.medhelp.org/land/sleep-tracker
I hope this information is helpful to you. I understand how troublesome daytime fatigue and sleepiness can be. Let me know if there's any other information I can get for you. Cheers, H.