I have CFS/M.E. as well. In fact, the diabetes was picked up when they did the barrage of tests to exclude everything else - there is no test for CFS, it's diagnosed by eliminating everything else it could be. I was also severely vitamin D deficient, but the huge supplements they gave me made no difference to the CFS. And if you really have got CFS/ M.E. you can't exercise through it. One of the key diagnostics is PEM - post exertional malaise. Any activity causes severe muscular pain, and a crash in energy levels, usually the next day. For some people graded exercise helps, but that has to start with tiny amounts and build up incredibly gradually. Even that is controversial because it's made a number of patients worse, even rendered them bedbound.
I struggle with LCHF and CFS/ME. I often don't have the energy to cook, and can only shop once a week. I try to make a stew, chicken or lamb with lots of vegetables and very little potato, once a week. That feeds me for 4 days, but even with a chopping machine it exhausts me. I made one yesterday, couldn't get out of bed this morning. Just chopping a few veggies causes acute pain across my shoulders and I can't stand for more than a few minutes. The rest of the week I eat omelette, fish, or tinned soup if too tired to cook. I eat porridge sometimes - less than 20 grams doesn't spike my sugars, but that's individual. I snack on cheese and nuts - too much cheese, and that has put my cholesterol up. Also eat dark chocolate, and I know too much of that. It's the one thing I haven't managed to give up.
I try to plan ahead so that energy goes into the important things like getting food and cooking it. But if I have a shower, I can't cook the same day. If I go to the supermarket I ride their in store scooters. But with CFS/ME, mental energy drains the battery as much as physical, so just driving around, avoiding all the people who ignore anybody not on two legs, uses up the energy for that day.
Since Christmas I have had a bad flare up of the CFS/ME, and a corresponding weight gain. Partly because the little exercise I was doing just walking round the house, I can't do. Partly because I struggle to cook and eat the right things. These two illnesses are hell together. And if you think the NHS is useless with diabetes, you should see what they're like with CFS/ME. Despite the WHO classifying it as a neurological disease, the NHS is still trying to use CBT for it. Having suffered from depression for over 30 years, and fought my way back from the brink over and over, held down a demanding full time job, and cared for elderly parents, I can categorically tell you there is nothing psychological about this illness, and you cannot think your way out of it. I know, I tried. And made myself so ill that I lost my job. The only thing that works for me is pacing and rest. Do a little....do nothing...do a little more.
Don't know if any of that helps, but hope so. Wishing you all the best dealing with these two illnesses.