I think you have assessed the problem correctly. The case is most probably the problem - if it is squeezed too tightly or bumped (in the right area of course) the case impinges on the pop-up and voila. I hope that you are right - a spring pop-up and a mechanical pull-down. One of the reasons that I am shy about prying the flash derives from my concern that the mechanical mech is somehow coupled and prying up will strip gears or some-such. I can see that you must be right about the spring pop-up - that relieves my concern about trashing a gear.
Okay, so this is the link that I mentioned in a prior post
I see that many comments are posted in support of adding exposure options but does anyone have any experience / knowledge of this site? If we enter a "petition" for instruction on how to fix this in the field, will Fuji listen?