Do bats use special relativity when they use echolocation?

Nope, the "$c$" in the Lorentz Transformations doesn't just apply to the speed of propagation of information in a medium, though I understand why you might think of it that way. Instead, the Lorentz Transformations are fundamental to the structure of space and time itself. In particular, they imply that the speed of light $c$ is frame independent: all inertial observers will agree on this quantity.

Such an argument cannot hold for the speed of sound in a medium, which is not something fundamental, but which depends on properties of the medium itself like its density and pressure. Indeed, objects can (and frequently do) travel faster than the speed of sound in a medium. Furthermore, it should be clear that if you consider just the bats moving in air, they don't even satisfy the basic criterion for relativity: for example, the Doppler Effect for sound clearly distinguishes cases where the "source" is really moving and where the "observer" is really moving (with respect to the ambient medium).

The big result of special relativity is that there is one -- and only one -- frame independent speed, and that is $c$, the speed at which light happens to travel. It makes no physical sense to speak of a speed greater than this value. The speed limit of 343 m/s in your problem, however, is a biological limit. If the bats in your problem were replaced by fighter jets, they would certainly be able to move faster than the speed of sound. So I see no reason to draw a parallel. In other words, you don't need to teach the bats Special Relativity: good old-fashioned Newtonian mechanics should work fine.

No. Lorentz transformation is a consequence of the postulate of special relativity which says that $c$ is the same in all inertial reference frames. Analogous statement is not true for sound.

If a bat moving at $1/3$ of the speed of sound along the $x$ axis measures in its frame of reference the speed of sound propagating away from it along the $x$ axis, it will obtain $2/3$ of the speed of sound measured by a bat stationary with respect to the atmosphere. This is a consequence of the fact that the speed of sound is fixed in the frame of reference in which the medium of propagation is at rest. Any bat moving with respect to the medium measures a different speed of sound depending on direction. Since the velocities involved are non-relativistic, the exact speed of sound observed can be computed using standard non-relativistic velocity addition rule.

By contrast, if a bat moving at $1/3$ of the speed of light along the $x$ axis measures in its frame of reference the speed of light propagating away from it along the $x$ axis, it will obtain the same result as any other bat. This is a consequence of the postulate of special relativity mentioned earlier.