There are several things to consider in the debate between realistic vs. indeterministic and Bohmian mechanics vs. no-go theorems (which includes the Leggett inequality). First of all, Bohmian mechanics is not a classical or alternative theory of quantum mechanics, but an interpretation of it. It reproduces what Schrödinger's equation and Born's rule predict, but gives us an explanation of why the universe seems to obey these laws. Second, the only ontological (i.e., real and measurable) property of the system (or the universe as a whole) is the position of the particle (Bell called it Beable to avoid over-interpretation of the term "particle"). Even measurements of spin, polarization and so on are then reduced to mere position measurements. This is called contextuality. To give an example of what this might look like, consider the Stern-Gerlach experiment: we shoot a beam of spin-$\frac{1}{2}$ particles in a directed magnetic field onto a screen and observe a splitting of the beam; we get two separate measurements of particles on the screen, one above the center line and one below it. One might be tempted to say that we have measured the ($z$-component of the) spin of the particle. But that's not correct! In fact, all we did was measure the position of the particle by having it hit our screen and seeing where it landed. Our equations and analysis now tell us that this is due to the particle's spin. But even then, you might ask, "Well, what's the difference?" The difference is that the particle never "carried" spin, the wavefunction did. The wavefunction, through the pilot-wave guiding equation, guided the particle according to the spin of the wavefunction; our measurement didn't naively reveal the spin of the particle, but through a positional measurement revealed the spin of the wavefunction.
So what does this mean for Bohmian mechanics vs. the Leggett inequality? The inequality (and all related inequalities) assume a rather generous notion of macrorealism, i.e., quote: "A macroscopic object, which has available to it two or more macroscopically distinct states, is at any given time in a definite one of those states." In short, an object is real if it has different possible states that it can be in uniquely at any given time. No mixtures, no superpositions, no collapse, etc. The problem now is that Bohmian mechanics doesn't give you "macroscopically distinct states", but positions. And every measurement is reduced to positions. One of the experiments that shows violation of the Leggett inequality uses polarization, and I even read in discussions of the inequality that "polarization measurements" (in the sense of naively revealing the polarization of the particle wave) refute Bohmian mechanics. But you cannot measure the polarization, you measure the position of the particle, which is guided by the polarization of the particle wave function. Bohmian mechanics does not satisfy the requirements of macrorealism (as put by Leggett and others), so a violation of the inequality does not rule out non-local realistic hidden variable theories (of which Bohmian mechanics is a part).
Hope this helps :)
