Given a set of observable facts, let's suppose that there are multiple hypotheses compatible with those facts that can explain them. Is there any way to tell which one of these hypotheses is ultimately true? Or are we doomed to use "rules of thumb" and heuristics to "rank" the hypotheses in an attempt to handle the uncertainty?
No, at least not given those assumptions. Given a set of facts, if we have several sets of hypotheses that each explains the facts, then there is no theoretical means to decide which set of hypotheses is true, if any, and which is false.
However, it is always possible, at least in principle, to falsify the hypotheses that are false by uncovering, or discovering, new facts.
This is what we do whenever we try to understand a problem. We imagine various hypotheses, and we try to falsify them. If this happens, then we have to imagine different hypotheses.
In a big research project, this produces a development cycle whereby we observe a part of the world, then we try to imagine plausible hypotheses, then we elaborate a theory based on them, and then we try to falsify the theory and therefore the hypotheses (or at least one of them) by uncovering some fact that does not fit with the theory. And then we start afresh by modifying our set of hypotheses to take into account any new facts. This process can potentially last for a very long time, possibly "for ever".
In effect, this is what science has been doing since humans have started to try to understand the world. Scientists still today try to make more precise measures of General Relativity effects in the real world, to see if the measures disagree with the theory. The process could potentially last as long as we are not sure that our science is true of the world.
If several theories could explain everything that could potentially be known, then these theories would have to be equivalent. However, I don't see that we could ever actually realise that we are in this situation. It is conceivable that we could have a theory that explains everything that could possibly be explained. If we ever have two different such theories, then we will be unable to decide which one is best. In effect, they would be absolutely equivalent.
This would not matter, however. What matters is that the theory allows us to predict the future, so to speak. Two equivalent theories would by definition produce exactly the same predictions. We could think of counting in base 10 and counting in base 2 for example. The results will be inevitably equivalent, even though the actual figures appearing in a base 10 count and a base 2 count will be different.