Yes; after each uplink you can send data back to the node. And when only sending a ping once per hour and only responding when the pet is lost, that seems reasonable.
But a pet can run quite far in an hour, so you’d have to do some calculations about how often you can send such ping, or implement some geofencing to only send pings when not at home or when not close to the pet owner’s Bluetooth-enabled mobile phone. Also beware that both the uplink and the downlink might not be received (especially for a moving node), so the application might need to repeat the downlink if it still doesn’t get location data after it sent a downlink earlier. So one ping per hour might really be too few for this use case.
As an aside: for technical reasons, using downlinks will really need to be limited. The TTN Fair Access Policy might only allow 10 confirmed uplinks and downlinks per day (and other operators should impose such limits as well; LoRaWAN is just not suited for downlinks very well). But I guess pets don’t get lost that often, so most pings would not need a response.
See also Is anyone already working on the PetFinder use case?