Random update ...
We sprayed the (dry) cat fountain components with Bitter Apple. This seems to have have the desired effect of restricting Whitey to drinking from it, as opposed to disassembling it.
When I got up this morning and went into the upstairs hall, Emily was in the penthouse of Tiger Tower again. She smoothly strolled down a couple of levels to be better petted. She was absolutely delighted when I ran the window blinds up to get her warm sunshine.
We remain worried about Emily, but not about her visits to Tiger Tower. She just plain likes it, and clearly can navigate its levels with ease.