Could the movement of a PBH found in the outer solar system by microlensing be totally unpredictable and impossible to track after it was first detected?
To determine the orbit of any body around the sun, at least three observations are needed at different times. This is true for black holes as much as it is true for asteroids or comets. With three accurate observations at three different times, a Keplarian orbit can be fitted.
The trouble with microlensing is that you only get a single observation. Even using additional clues (such as the length and brightness of the microlensing, from it's inferred effects on TNOs, from the observation that it doesn't seem to enter the inner solar system) you don't have enough information from a single microlensing event to determine the orbit.
The orbit would not be "random". It would still be an ellipse, with the sun at the focus, and some elliptical orbits might be more likely than others. Still, the orbit would not be determined by a single microlensing observation