Re: having had a "mystical" experience
I don't know why you object to this. It's not an error, and is quite common when referring to something experienced in the past.
I recall having had a delicous meal when passing through this town many years ago.
She remembered having had a bad feeling when she first met him but ignoring her intuition.
I suppose the meaning is the same as "I recall that I had had" or "She remmebered that she had had" but you don't have to repeat the subject with the "having had" construction.
40 percent report that they had had = 40 percent report having had.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.