OK, I finally got around to trying this again with a better test scene and more controlled conditions. First is the control case: a single exposure with a single flash into a small umbrella to camera left and up a bit, no fill flash or reflector. This was metered at f6.3, ISO 800, but I found an exposure of f9.5 gave the apples a better colour, and I wanted deep shadows to try to rescue, so went with that. Straight scan with no manipulation.
Control case: single exposure by
Howard Sandler, on Flickr
I then repeated with another sheet from the same pack, same main exposure. I followed this up with a post-exposure of an "approximately" middle grey sheet of paper (same as the background sheet) at f16; i.e. 1.5 stops down from the main exposure, or 2.5 stops down from the metered exposure.
Post-exposure case: exposed scene, then grey sheet by
Howard Sandler, on Flickr
I can see a bit more colour in the dark green leaf in the foreground that hangs over the grey card, and slightly more detail in the dark orchid petals, but overall, the black point is slightly lifted as well. I did try another post-exposure a bit less, at f20, but didn't see any effect, so I have not bothered to upload the scan.
I conclude that pre- or post-exposure (I haven't seen any evidence the order of which exposure comes first matters) exists as a phenomenon, but is too fiddly with Instax to bother in most cases. The range between no effect and muddying the shadows is too fine.