Praise be to Allah.
If the situation of this poor person is as described in the question, then he should pay the saa’ of food on behalf of himself, if it is surplus to his needs and the needs of those whom he is supporting on the day and night of ‘Eid. This is because the Prophet (peace and blessings of Allah be upon him) said: “Start with yourself and then with your dependents.” (Narrated by al-Bukhari, 2/117, 6/190; Muslim, 2/717, 718, 721, no. 1034, 1036, 1042)
With regard to his dependents, if they do not have anything to give as zakat al-fitr on behalf of themselves, then they are not obliged to pay it, because Allah says (interpretation of the meaning):
“Allah burdens not a person beyond his scope.” [al-Baqarah 2:286]
And the Prophet (peace and blessings of Allah be upon him) said: “There is no sadaqah except for one who has the means.” (Narrated by al-Bukhari, 2/117, 6/190; Muslim, 2/717, no. 1034)
And he (peace and blessings of Allah be upon him) said: “If I command you to do a thing, then do as much of it as you can.”
And Allah is the Source of strength. May Allah send blessings and peace upon our Prophet Muhammad and his family and Companions.
And Allah knows best.