I once watched an online seminar by Robert Spekkens, who said something along the lines that no-go theorems are interesting, because they put constraints on what an epistemic interpretation of quantum mechanics can look like. Any no-go theorem makes a certain set of assumptions, and if the theorem is correct, we know that we must avoid at least one of those assumptions if we're to make a successful theory.
The Pusey-Barrett-Rudolph paper spells out some of its assumptions. (They do it most explicitly in the concluding paragraphs.) There may well be additional unmentioned assumptions (e.g. causality), but the ones they specifically mention are:
There is an objective physical state $\lambda$ for any quantum system
There exists some $\lambda'$ that can be shared between some pair of distinct quantum states $\psi_1$ and $\psi_2$. That is, $p(\lambda=\lambda' | \psi=\psi_1)$ and $p(\lambda=\lambda' | \psi=\psi_2)$ are both non-zero. (This is what it means for an interpretation to be epistemic, according to Spekkens' definition.)
The outcomes of measurements depend only on $\lambda$ and the settings of the measurement apparatus (though there can be stochasticity as well)
Spatially separated systems prepared independently have separate and independent $\lambda$'s.
It is from these that they derive a contradiction. Any theory that fails to make all of these same assumptions is unaffected by their result.
I'm fairly sure that under the standard formulation of Bohmian mechanics violates the second one, i.e. Bohmian mechanics is not an epistemic theory in the sense of Spekkens. This is because in Bohmian mechanics the physical state $\lambda$ consists of both the particle's real position and the "quantum potential", and the latter is in a one-to-one relationship with the quantum state.
If I understand correctly, you're suggesting that you could instead think of the physical state, $\lambda$, as consisting only of the positions and velocities of the particles, with the non-local potential being considered part of the equations of motion instead. But in this case, the third assumption above is violated, because you still need to know the potential, in addition to $\lambda$, in order to predict the outcomes of measurements. Since this formation would violate the assumptions of Pusey, Barrett and Rudolph's argument, their result would not apply to it.
In your updated question, you clarify that your suggestion is to fix the wavefunction to a particular value. In that case it's surely true that Bohmian mechanics reduces to particles moving according to deterministic but non-local equations of motion. But then you have only a partial model of quantum mechanics, because you can no longer say anything about what happens if you change the wavefunction. My strong suspicion is that if you take this approach, you will end up with an epistemic model, but it will be a model of only a restricted subset of quantum mechanics, and this will result in Pusey et al.'s result not being applicable.
I think that 't Hooft's ideas about superdeterminism and Bell's theorem are relevant to this topic. If the universe is superdeterministic so that all experiments are determined by initial conditions, then the contra-factual arguments that lead to the Bell non-locality conclusions are ruled out. The universe only plays once - is how some have put this. In the early days, this was called "conspiracy against the experimenter", and it was categorically excluded from the discussions. It still remains the biggest loophole, and I don't think it can ever be eliminated. But it means that strict free will doesn't exist. The fact is that both Alice and Bob consist of a finite number of atoms each, and are certainly quantum mechanical systems themselves. Consider the bouncing drop experiments. It's surely a deterministic classical system to a high degree of accuracy. To mimic quantum mechanics, we must not only have bouncing drops that are entangled, we must also have measuring apparatus and observers which consist only of bouncing drops. We are super-observers of the drop motions, but we are not built up of bouncing drops. We are allowed to measure the drops without disturbing them, but "embedded observers" made up entirely of bouncing drops may not be able to measure things as finely as we can. To those embedded observers, the universe of the bouncing drops might appear as undeniably entangled. So there is hope for the bouncing drops as an example of 't Hooft's superdeterministic realization of quantum mechanics as an emergent phenomena. I have no idea whether the classical lagranians for the bouncing drops will allow this behavior, but I see it as the only hope for a description of entanglement in this classical framework.
Best Answer
The "pilot wave" is the same as the multi-particle wavefunction in quantum mechanics, so it evolves according to the Schrodinger equation. Such a wavefunction is not a field in three-dimensional space, it is a field in 3N-dimensional space where N is the number of particles.
The equation of motion for the particles in three-dimensional space depends on the gradient of the complex phase in that 3N-dimensional space. This part is exactly the same as classical Hamilton-Jacobi theory, which is an alternative representation of the forces of classical mechanics. But building trajectories from the quantum-mechanical wavefunction adds an extra nonlocal force.
You can't send nonlocal signals in Bohmian mechanics because you can't do that in quantum mechanics, and Bohmian mechanics is just quantum mechanics plus particle trajectories. The reason you can't do it in quantum mechanics is because the nonlocal correlations it induces aren't strong enough, they are only strong enough to add a nonlocal extra to a local signal (see quantum teleportation, which requires a local signal to be performed).