Thank you for pointing out the obvious flaws in these scenarios. Maybe replace them with this:
Grudge the cat gets lost in Discovery's turboshafts. Realizing the search could take decades, Burnham makes a very emotional appeal to Book (complete with tears and hugging) to cut his losses and get a new cat. But Book never gives up, and there's a 1 minute scene in every episode for the next eight seasons of Book wandering through the turboshafts looking for the cat. Eventually, he finds the cat and there is a very touching reunion scene.