I'm unclear about which subgroup your theory fits, but it doesn't seem to fit all subgroups. For me, physically-induced PEM has a consistent ~24 hr delay. Other people have much shorter delays. See how that fits your theory.
Also, I've taken prednisone, and it didn't trigger extra fatigue. The first two times, it triggered temporary remission. I don't know whether the immune system would differentiate between cortisol and prednisone. Prednisone does reduce production of cortisol, but I don't think a small reduction in cortisol would cause a significant reduction in autoimmune response.
Do your symptoms rise and fall with the cortisol cycle? My PEM symptoms don't seem any worse in the morning (peak cortisol). My overall ME symptoms get worse in the afternoon.
Try thinking your theory through with timing in mind: cortisol peaks, autoimmune response times, etc. You might find that it fits, or you might find an obvious flaw.
Another thing to keep in mind: temporary remission. Does your theory explain why an ME victim can, over the space of minutes, switch back to full health, stay that way for hours, and then switch back to ME? I don't think autoimmune reactions work that way, but I'm not an expert.
Personally, I don't think ME is an autoimmune response. I think there's something interfering with the 'return to normal' state after an immune response.