Your problem will be the weight/transmission distance ratio, you don't get anything for nothing. And the cat a stiff neck.
Right, which is why I'm focusing on a manual locating approach (go look for the cat) rather than automatic tracking (sit at home looking at a screen). I can live without always knowing where the cat goes, but I would love to have a tool to help look for the cat when/if it does not get home at night. From those requirements, there should be no weight/transmission issue - currently looking for a way to emit beacon pulses onto the 457KHz band (official avalanche victim transponder frequency*) and I reckon a 8mW 200ms pulse at 1Hz would run a while on a 3.7V 100mAh (370mWh) LiPo weighing just 2-3g.
PS: I live in Denmark where the highest "mountain" is 150m and the most amount of snow is 20-30cm so I am extremely unlikely to interfere with real avalanche distress scenarios.