What is the irony of Félicité's long search for Loulou?

When Loulou gets lost, Félicité searches for him until she is entirely worn out and exhausted. After all that, the irony is that he returns home to her of his own accord after she has given up. Her love drives her to great extremes to keep him and protect him, when he is actually fine and wants to return to her. The search itself is ironic, but it is also ironic that, when she finally finds reciprocal love, it is in the love of a bird.