I think you should ask for a redo of the LEMS blood test. The pattern of "okay at first, worse after some exercise and then better" is classic LEMS.
Did you ever have migraines during times when you didn't have a period? Simply because something makes a migraine worse does not mean that it is the cause. It could be an aggravating factor. You can't separate the CNS from the endocrine system. They are completely interrelated. In fact, some researches are now calling it the neuroendocrine system.
Yeah, periods can increase fluid retention but that whole system is related.

More hormones, more fluid. But that doesn't happen in everyone (especially not men).
Well, I have no idea what is going on but there are too many signs that would make a good researcher want to nab you and not let you go until they figured it out!
Some CMS's are also like what you described with your doctor's exam. They don't fatigue as easily but when they do, they go downhill fast!!! Then they rest and recover. You won't like the RNS they normally do for that. It's a RNS for ten minutes. Or as long as you can take it. I've had it and the doc was amazed I lasted as long as I did. But this is why you need a CMS expert! They can usually figure all of this out. Read the PDF. It's an interview done about CMS with UC-Davis. I'm glad I kept it.
I hope you have a great conversation with your neuro. I'm sure he wouldn't want to miss anything. Make sure you reassure him, especially about the Imuran. If this ends up not being autoimmune, he'll want to know that he didn't "harm" you in any way.
I love a good mystery. I hope this helps you too, Mike or anyone else battling these questions.
Annie