Most cheap ultrasonic transducers are tuned for a narrow resonant frequency, with 40kHz being the most common one. I've not tested one at other frequencies, but it'll be similar to one of those piezo discs. A small electrect microphone will be much better.
Using a frequency counter is a bad idea, because it rejects all the amplitude information. It will just lock on to one sound, making it incapable of detecting more than one bat. The only advantage is it scales down the entire spectrum.
I looked at the kit schematic again. Attached is just one channel, with the component values made clearer, as the picture in the PDF was low resolution and very fuzzy.
The first two op-amps amplify the signal by a factor of 121, with a lower cut-off of 16kHz. The third op-amp then inverts the signal. The analogue swithers are alternately switched on by the oscillator, selecting between the inverted and non-inverted signal, at the oscillator frequency. This is then low pass filtered by R5 and C11, with a cut-off of 7kHz.
I think I might use a similar circuit, but it can be greatly simplified. Rather than using an op-amp to invert the signal, just have one op-amp, which can be programmed to inverting or non-inverting, with an analogue switch. This doesn't cut down on the number of op-amps but it means the oscillator doesn't need an inverting output and only one analogue switch is required.
I simulated it and the results are decent. R4, R5 and C1 bias the signal to half the supply voltage. When S1 is on, the circuit is inverting and when off it's non-inverting.
I'll swap the final op-amp for an LM386 to drive a small speaker.
I might also use a common emitter amplifier for the first microphone amplifier stage, because it should have lower noise than the TL072.
EDIT:
I've simulated the microphone amplifier. The response probably counteracts the reduction in sensitivity vs increasing frequency of the electret microphone quite well.
I'm moving towards using a TL074, with two sections for the amplifier, one for the inverting/non-inverting amplifier part for the heterodyne circuit and one as Schmitt trigger oscillator.
EDIT:
For completeness, here's the BJT amplifier from the website linked below, which uses the obsolete NE612 as a demodulator.
http://bertrik.sikken.nl/bat/ne612het.htmThe microphone is noddled as a signal source with an impedance of 4k7.
It has a sharper peak, than the op-amp circui9t, but lower noise. I might use two stages, a BJT and op-amp for more gain.