Bohmian mechanics will predict something different *if* you allow the possibility that the particle distribution at some time does *not* conform to the Born rule. The reason this is hypothesized is that there is a theorem that suggests that if you start with a non-Born distribution, then over time it will gradually evolve, in a manner akin to thermalization in thermodynamics, to a Born rule distribution asymptotically, and it will do so with probability 1. And thus if you allow for this, it makes the Born rule able to be a derivable theorem, as opposed to a postulate that has to be assumed from the beginning, and thus makes the theory slightly simpler. Such a deviation from Born behavior could be imagined to have been at the Big Bang, and this has been suggested would lead to possible observable consequences in cosmological studies for the structure of the early Universe and thus of remote areas such as the CMB radiation. However, failure to observe this would not falsify Bohmian mechanics, rather merely those versions that posit non-Born behavior at the Big Bang. We could still assume the Born rule as a postulate that has held from the beginning, and then there is no refutation.
How does this jive with the issue of interpretation versus theory? Simple. The Born rule is part of standard quantum mechanics. If you remove the rule to derive it later as a theorem where you start the hidden parameter with a non-Born distribution, then you are no longer simply interpreting QM, you are creating a new theory, as you have modified the original theory by removing a component as a basic element and instead substituted in that role a different element (the hidden positions) from which you are going to derive it. If, however, you do not do this, then you do indeed have an interpretation: no prediction it makes is different from that made by standard QM. You have added a new element, but you have not modified any of the rest, and the new element admits of no contradiction, and simply serves to fill a philosophical gap.
The only reason this modification is considered is that it looks attractive since the theorem of approach to Born behavior in the limit is mathematically proven, and thus it is tempting to go there as fewer assumed principles in a theory is a nice thing to have, but it is not at all necessary.
That said, if we DID observe the variations predicted by the non-Born distribution hypothesis, then we would know that Bohmian mechanics is the "correct" way to understand QM, and moreover, standard QM is falsified as a strict theory.
If anything, the real difficulty with Bohmian mechanics is that it really only works naturally for nonrelativistic particle QM (and so also, the appearance of faster than light speeds in it should not be taken as a strike against, it IS formulated on a Galilean space-time, not a Lorentzian one, where that infinite speed is allowed, just as in Newtonian mechanics!). It doesn't straightforwardly generalize itself to the case of relativistic quantum field theories which we know are the more broadly-applicable ones, and moreover it does not provide a general framework for interpreting an arbitrary quantum theory. However, this is actually not something that should be taken as a surprise: It is often forgotten, but Bohm actually technically did not intend for this interpretation to be a final one. He simply gave it as an example to show that you *could* make a theory in which a classical-like, deterministic reality could exist between measurements, and have it reproduce all the observable predictions of standard QM, contrary to some detractors at the time (and some which still exist now), and in that, he was absolutely correct. Thus its non-generalizability should not be a surprise because it was not created for that purpose to begin with. It was a proof of concept, so to speak.