HAve you actually started flying yet?
I ask because this statement "but running and hiking can make me feel a little bit like I'm dying - lightheadedness, heart working overtime, pins and needles in my limbs. Major postural changes like going from crouching to standing make me dizzy too. I'm fine once they pump me full of the good stuff of course"
is a little worrying. Flying can be a lot more physical than you think ( at least I find it so) and I would not be flying with Anaemia so bad that exercise "makes you feel like you are dying"
not a doctor but someone who was suffering from anaemia similar to yours for a very long time. Luckily for me the cause was very obvious and eventually sorted out surgically.
Personally I found that although I didn't realise it at the time, it affected my cognitive processes massively as well
You really shouldn't be flying with Anaemia bad enough o cause physical symptoms like the ones you mention.
I hope you get some answers soon