How would a small nearby black hole be detected and confirmed as such?
I would say that the only way this can be done is making ever more careful observations of the TNOs that are claimed to be affected by this putative black hole.

Ultimately such observations should narrow down the location of the object. Even then, confirming it's existence will be very difficult. It will have a small lensing/magnification effect on background stars, but it would have to get very close (in terms of angular separation) to a background star for this to be observable. It's unlikely that the position could be pinpointed accurately enough to predict when such a microlensing event would take place.

Perhaps a better bet would be to send a probe, like New Horizons, to its likely location, where the gravitational effects could be studied in situ.