Do echo-locating bats experience Terrell effect?

I don't think it would happen at all - Terrell rotation is specifically due to the Lorentz contraction from special relativity, not just from the Doppler effect (which, in any case, takes a different mathematical form in the nonrelativistic case).

Most common bats usually fly at a rather sedate 16 km/h or so, so they wouldn't experience much of a Doppler shift. But the fastest bat ever recorded has flown at 160 km/h or 13% of the speed of sound, so Doppler effects should indeed be quite noticeable (after all, you can hear the Doppler shift of ambulance and police sirens when they're just moving at ~80 km/h). In fact, bats actually adjust their echolocation pitch based on their flight speed to compensate for the Doppler effect in order to keep the return pulse's frequency within their optimal hearing range.

Extremely rarely, one finds bats capable of even more complicated radar data processing. But I don't think they would "see" a Terrell rotation.

I've read paper of R.Penrose linked on wiki page (in fact as it is on subscription basis only first page is available). research note by N.Penrose. It says:

the light from the trailing part reaches the observer from behind the sphere, which it can do since the sphere is continuously moving out of its way

So effect is neither due to Lorentz contraction nor Doppler shift, but from finite speed of light and high (relative to that speed) movement of object relatively to the observer. As other answer already noted speed of bats (relative to sound), the Terrell rotation is possible to experience for bats.

ADDED: after some hard thought I keep final conclusion: Terrell rotation in some form is possible for some settings (distance, angle and relative speed) for echo-location of bats. However I realized two key differences for sound echo-location from original effect: 1) light is relativistic, whereas for sound in air whether object or observer moves matters 2) active echo-location means bat "hears" reflected sound it emitted, whereas for light observer is assumed to see light created by other objects. So time for sound to travel from the bat to the object had to be taken into account to calculate actual "image" of the object.

"Hearing" response from "not seen" from initial position part of the object is possible because sound wave can "go around the corner". Then due the fact that the bat changed position, bat would hear response from that part simultaneously with response from "front" part, therefore experiencing effect similar to Terrell rotation.