If you could excite this with a pulse of the appropriate frequency, and if it were nearby, you might receive a slight signal back from the oscillating element after the exciting signal cut off. This signal would have a characteristic amplitude decay over time which should be easy to characterize from experiment with similar pingers. You would also receive echos of the original pulse as it bounced off more distant items, but these would not have the characteristic decay signature.
Oh wow! Have you been reading your "Magnetic Resonance Imaging for fun and prophet" Machinbird? That is exactly the basis of MRI and you may have hit on a novel and
just possible answer. Nice one.
The return signal will, of course, be tiny and distance will work against you dreadfully, but at the frequencies we are talking about, powerful transmitters are already available and suitably sensitive receivers too. Experimentation would, as you suggest, be necessary to study the characteristics of the received FID (Free Induction Decay) and what proportions of transmit and receive would be necessary.
And you wouldn't need a nice, clear signal ready for image reconstruction, just a return - any return - showing the FID of the pinger.
Very nice lateral thinking sir, if I might say so?