The postural hyperventilation (POSH) is related to dysautonomia, not autoimmune disorders, as we see clinically it happen alongside POTS and breathing is controlled by your autonomic nervous system. It isn't shortness of breath as in asthma, it is a literal increase in your breaths per minute and decrease of eTCO2 upon standing only. It reverts almost immediately upon laying horizontal. See this article
I also have some light asthma a good portion of my life from cats and I've been on steroid inhalers for 7 years before I got ME through to today (8 years total). ME seems to have worsened it a bit, but overall it's separate from my ME. I also have Crohn's disease. What issues specifically do you think are autoimmune? My C-Reactive Protein, ANA, and Sed Rate were tested multiple times and they all are fine repeatedly.
I'd definitely be interested in more thorough autonomic testing. I will try to have that done as that could be helpful.
I think this type of thinking (I didn't have it, so testing isn't necessary) likely won't get us far with ME. I think we need to encourage doctors to cast a wide net, testing for more, not less, to capture all component issues of ME a patient is dealing with. Painting a fuller picture of the pathology (especially related to dysautonomia which is neurological) both individually and for the disease as a whole would lead to progress in our understanding of the disease.