Personally, if I were toying with this application, I'd start out with something a little slower. If you used ultrasound instead of radio waves for ranging it would make life much easier. I'm not talking about the kind of echolocation commonly played with, instead building a US transceiver. Your stationary device (the human) could then broadcast a time stamp via radio waves for synchronizing and timing the US handshake.
So, cat sends an ultrasound packet encapsulating a timestamp received on its RF radio. Human receives packet and calculates range based on the fact that it sent the radio timestamp.
Since the speed of sound is approximately 1 ft/ms as opposed to 1 ft/ns for rf, you have a snowball's chance of measuring flight path with the kind of technology you're talking about using.