How about this:
1. The bird does it's thing and flies off screen
2. Instead of destroying it, just suspend it (disable it's behavior, there are various ways to do this: e.g. disable a group, set a boolean to false or a state machine) use the timer behavior to start a two second timer on the bird
3. On timer, reset & enable the behavior of the bird again (as in give it back the rock) and set its position to Y = whatever height you want it to patrol and X = player.X + screen width/2 + some offset so it starts off-screen
4. This would also be the place were you can test if the bird spawns outside of its patrol area (if player.X + screen width/2 + some offset > rightWall.X) If that's the case, you can instead just start the timer again, which would mean every 2 seconds the bird will check if the player is in the right position for the bird to spawn inside the patrol area.