Having a saliva cortisol test is understandable in your situation, I've had it myself as has practically everyone else on this forum I imagine.
However....what I think you need to consider is the following.
1) An endocrinologist would consider a saliva test a bit 'DIY', in that it's the first step to then requiring 'gold standard' tests that only then usually then require medical intervention with drugs. By that I mean it's unlikely you'll find a doctor who will rely on a saliva cortisol test alone, and instead they'll stick to the gold standard test methods. Although expensive, I would go down this route as it's far safer.
2) If you don't and a doctor will prescribe you drugs like candy, be aware that taking hormones may thus have some undesirable or even seriously dangerous effects. Years ago, it was thought that the hypothalamic dysfunction in 'CFS' could be fixed simply by boosting cortisol. Short term this worked for some, but not long term. My unproven theory, is that by taking hormones you ramp up the infections, as the infections feed off cortisol hence the body lowers it as protective measure. Certainly if you look in medicine generally they don't like you to take hormones if you have an active infection, even something as innocent sounding as taking Thyroxine for low thyroid can help 'wake up' chronic infections. So take that on board a bit, that the idea that if you have ME CFS you can just 'fix' your cortisol issues (if you don't have another illness of course), is rather unlikely.
Before your doctors all give up, you have to have had these tests in my view to be absolutely sure the diagnosis is correct because as you know ME CFS is all guess work, there's no test. You can have 10,000 private tests, but even then no one 100% knows they have ME CFS. So we need to go back to basics and rule out conditions that can be diagnosed reliably. In this respects in comparison to immunology, endocrinology is child's play.
24hr Urine Cortisol at home and then give into the hospital the next day.
Short Synacthen Test in Hospital.
5hr Glucose Tolerance Test in Hospital
Cost wise, obviously 24hr urine test first (must be 24hr) and then do a Short Synacthen test next. The Glucose one is just to rule out hypoglycemia which can be linked to cortisol failure or other autoimmune issues affecting your brain or adrenal glands. Also take on board that very poor nutrition can screw up your body, and again although expensive, I would make sure you aren't squished in bed as you're running hugely low Vitamin B6/E/D/Zinc etc. Sounds far fetched, but on private testing, many people here with ME CFS, find out they are low in many different things they never dreamed off as they eat well, or even take supplements - but still, levels are low, (or far too high).
If you're bed ridden your local Hospital should take you to the Hospital by ambulance to have a test in the day unit. You want somewhere as quiet and stress free as possible.(Rather silly statement I know when you feel anxious and awful), but I mean don't do it in an ER or surrounded by people screaming and yelling at each other in an admissions/diagnostic ward. What I do, is I hire a paramedic crew and I befriend them. I do this, psychologically, to try and lower my stress/panic levels. This way, although my tests will still be falsely high, (stress increases cortisol and prolactin) and strong anxiety will increase adrenaline - by having people who are 'friends' with you, your brain can learn to relax a little more than going from a quiet house to a madness of a busy hospital. So I always ask for a medical day unit, which is usually nice and quiet. Better still, go privately and always tell the staff beforehand all of your medical issues. All of these things give you confidence, and help lower cortisol and other stress hormones - what you want! I've had to do this myself as I can't just 'turn up' in out patients and sit upright like everyone else.
Generally, in ME it is the brain not the adrenal glands that is deficient in producing cortisol (a neurological problem) not the adrenal glands (an organ problem). In this respect I would not be surprised to learn if your diagnosis is correct (you say you've only been ill 1.5 years) that your 24hr tests are low normal, but not low enough for Addison's disease which comes with a whole host of signs and symptoms you haven't mentioned thankfully.
That's not to say that you couldn't have ME + actual adrenal failure of some sort, or adrenal issues and not ME at all, but that it's more likely that you have a neurological problem if you have ME.
If this is the case the following is very important as already listed above:
A Short Synacthen test stimulates the adrenal glands, it does not work on the brain in the same way as other 'stimulation tests' that measure Cortisol. In this respect, to rule out pituitary dysfunction you would need a similar test, in which the pituitary is 'stressed' (e.g. a Glucagon Stimulation test), I've had this, and they measure you pre, post injection of a drug (whilst fasting). I've had it myself, it makes you feel super rough but you're safe and the test is only done with specialist nurses who are aware of any possible intervention. I would not agree to have this done outside of a hospital, or a specialist clinic. Make sure you are extra careful when having tests that are designed to 'provoke' a stress reaction as your mind cannot overcome your body, if your body doesn't work correctly - as we all know.
If your brain does work correctly and you have another reason for having ME symptoms most likely, your cortisol after the injection will remain 'up', after initially high (from the adrenaline rush of anxiety/panic over the test which is due to your adrenaline, coming from your adrenal glands which is good if it does this!).
However, in ME which usually involves an element of 'central' hypocortisolemia because your HPA axis is dysfunctional, you will see that your levels of cortisol after the injection do not rise sufficiently, e.g. mine was something like 440, and the minimum it should be is 550. BUT...this is then a diagnostic nightmare, because they can't really do anything for you in this situation as your neurological problem (how your brain reacts to stress, due to hypothalamic dysfunction) cannot be fixed simply by taking tablets. Catch 22. In fact in 'CFS' past studies found giving patients hydrocortisone tablets makes them worse.
Good luck with your test, but if you have ME, try and be realistic that the results may well be normal, or low normal which means (usually) and endocrinologist will lose interest in you as they are used to people with hardcore adrenal failure, adenoma's that need surgery or even brain tumours, not more 'subtle' neurological problems mostly caused by the immune system and autonomic dysfunction - hence gross endocrine problems are rare in ME.
Best Option:
If you THINK you have ME but aren't sure, I would definitely rule out Adult Growth Hormone deficiency first which is rarely done for anyone with ME CFS. IGF-1 blood test is not sufficient to rule this out. You need a full battery of pituitary tests, a pituitary high resolution brain MRI (if the bloods are abnormal) and ideally the Glucagon Stimulation test to measure post stimulation Growth Hormone or similar to 'stress' your pituitary where they can measure Cortisol, Glucose, ACTH and Growth Hormone all at once. Adult Growth Hormone deficiency can mimic lots of symptoms of 'CFS', including shortness of breath, low blood sugar events, anxiety/agorophobia, weight gain, depression, breast growth in males, heart problems, cognitive dysfunction, the list is endless.
Alternative:
If you cannot get this test due to cost, or too ill. Try to source a 24hr Urine Growth Hormone test. You must sleep correctly, or the result will be falsely low. In your sleep, your pituitary releases little bursts of growth hormone. So for this test, I dehydrate myself (otherwise I pee all night and if I pee I'm awake and not sleeping, with no sleep I won't produce GH). If this is low, I would repeat the test. Again, make sure it's fair and you aren't laying their wide awake.
So if the test is legitimate, and low, then you can go back to your doctor and say look - my GH is really low, please can I now have the GH measured in a Hospital based Stimulation test?
See how that goes and good luck.