Arise,
These are just personal thoughts based on my own experience with both NHS and private doctors over the past six years or so since I was diagnosed.
Going back to your main points...
The ME diagnosis is helpful for practical things like disability, insurance etc. and to rule out certain conditions that may generate similar symptoms such as for e.g. celiac or certain autoimmune conditions. Personally, I wouldn't waste my time with GP other than ask him to refer you to an appropriate consultant for a proper assessment. For that, it's easier for you to go places where consultants do not believe CFS is psychological. ME charities etc can help you find one local. The subject of MS is probably better raised with a consultant and asking for tests that may rule out the possibility (I have no idea what the process would entail).
As for getting better, or healing, this is hardest part.
If your diagnosis is going to be ME, the NHS won't be able to help. And there's more bad news I'm afraid.
At present, I do not believe there's a single doctor in the UK good enough to surely help you with that, unless you're very lucky (almost always an oxymoron with ME).
I'm referring to private doctors of course. They are very few, have their own ideas and bias in approaching this illness and a certain inflexibility which I have personally found unhelpful (in those I have personally met).
In terms of tests that may be helpful in understanding what's wrong with your health, again this is my personal view, but if what caused your illness was an infection, and judging from your symptoms list that appears plausible, then that's where the focus should be. Sounds logical, no? Not according to many doctors, would you believe it.
The stool/gut test will probably be helpful for highlighting any yeast or bacterial overgrowth or parasites. If so that's something you can treat.
In addition, because you have been ill for some time you're likely to be overburdened with toxins and getting your system able to rid of those should be a priority.
I'm not a doctor but of all the tests that your doctor might include, personally, I would insist on the following: infections panel (including Lyme western blot), methylation panel, gut/stool test, hair analysis and stool test for toxic metals, nagalase.
Regarding your Vitamin D, I'd ask for a test of both 25-hydroxy-vitamin D and 1,25-dihydroxy-vitamin D. Very few doctors test for both but it can help determine certain possible causes for your low 25-D. I personally wouldn't supplement until that's clarified.
I wouldn't bother with mitochondrial testing, genomic panels and most nutritional profiles. I did all of such and found them not helpful in managing effective treatment, as my illness was also triggered by an infection.
They're usually requested by doctors that aren't able or willing to distinguish between the illness key drivers and secondary effects which are usually endless.
In order to get better you'll need to focus on the first ones whilst avoiding getting sidetracked chasing the others.
This is the hardest part and for this you'll need an exceptional doctor.
All the best