We're talking about 200 years before the thing is even 1000 AU away. Even 100 years from now it will probably be comparatively trivial to send out a probe to vi

We're talking about 200 years before the thing is even 1000 AU away. Even 100 years from now it will probably be comparatively trivial to send out a probe to visit the asteroid, let alone 200. Also, interstellar asteroids might be comparatively more common than we've seen so far, but given the basic factors of how widely separated stellar systems are it's quite likely that they are still very rare. So even if we have discovered many more interstellar objects traipsing through our Solar System over the next many decades I highly doubt we'll have so many that we wouldn't want to visit all of them.