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.