With increasing dynamical evidence, we should be able to get every better localization and mass for the hypothetical object. With an accurate enough read it should become possible to rule out a conventional planet-like object by lack of direct observations at the predicted location.
However, this will not necessarily tell us that the object in question is a black hole. For example, a planet made of dark matter would present in almost the same way (and would also be exciting!).
In situ observations by an orbiting probe would be needed to further determine the nature of the object.
Just by orbiting the object we would be able to learn a lot. For example, the radius of the orbit will allow us to get a lower bound on the density of the object. By steadily shrinking the orbit one could, rule out many alternatives. However, there will be a practical limit to how small we can make the orbit before a probe is torn apart by tidal forces (this would happen long before reaching the innermost stable).
However, once you are this close to the hypothetical black hole there are other experiments you could do. For example, black holes have characteristic frequencies (quasinormal modes) with which they oscillate if you excite them, for example, by scattering off a wave. For a black hole of several Earth masses, the electromagnetic quasinormal modes fall in the microwave regime. By scattering microwaves off the object and looking for the quasinormal modes we can learn a lot about about the nature of the object. (Observation of the qnm spectrum will tell us that it is a black hole, if it is an ultra compact object we would expect to see echoing bursts, etc.)