We’re pointing the JWT pretty far out, so why wouldn’t we just point a camera at this thing?
Also it's predicted that a star comes with 1,500 AU of the sun every billion years or so, a star could have thrown things out of whack and left.
But if you suspect it’s a planet, you know it’s in orbit around the sun. Feels like something we can track.
Our planets' orbit do tend to be somehow aligned, so you could make the (possibly wrong) assumption that the 9th planet is also somehow aligned. This might reduce the search space by 50%+.... but even then you'd still be several orders of magnitude off by the precision needed to go looking for it.