Let’s try a different sentence: “Can a cruise ship be used for passengers after having been/being sunk?”
I would say, the perfect case, “having been,” is called for. If the vessel were still on the ocean floor, obviously, it could not carry passengers. It was sunk, and then re-floated and presumably refurbished. Its unfortunate status, its sunk-ness, is in the past; it is “perfected” in the grammatical sense.
With an abandoned animal, it too has been un-abandoned somehow; it has been adopted or rescued or re-homed, so we should say it “had been abandoned”.
However, the point of the sentence is that, perhaps, being abandoned is like being orphaned: it never really goes away. Once abandoned, the animal will always being alienated from humankind to some extant.
This difference is subtle, at best, and I think a lot of native English speakers would not notice it.