Watch the skies
As far as speculating about the Rama Scenario goes, even if we can't visit, it's likely the trajectory will yield up some clues. Changes, of course, would be an obvious smoking cigargun. But there don't have to be course corrections for the object to still be intentionally on a slingshot trajectory. "They" could be just as good as JPL at that maneuver.
What I want to know is, is there anything along that extrapolated trajectory within a few light years? Is there a reason to suspect an opportunistic use of the Sun, versus an accidental hyperbola?