I should, as food is likely to grow scarcer due to this disaster event.

[Byleth literally went two solid weeks without food once during one of the harsher winters in Fódlan (as did Jeralt), and the only thing that happened was that he felt kind of nauseous by the end of it (whereas Jeralt had been slightly worse off). He had his provisions. If he spaced it out, he could last for probably an entire month without needing to take any food circulating the city's provisions.]

You and Claude will require at least two meals a day to maintain your strength at optimal levels. I can manage the same with only one meal every three days. Therefore, it's logical for me to fast to allow others requiring more food to maintain their strength for longer.