Is the head of a poor household who possesses an amount deposited in an Islamic bank, and fears that withdrawing a portion of it to perform the Hajj obligation would affect his monthly income, required to perform Hajj?

If your situation is as you described, you are not obligated to perform Hajj due to the lack of legitimate ability, as Allah Almighty says: "And [due] to Allah from the people is a pilgrimage to the House - for whoever is able to find thereto a way," and His saying: "So fear Allah as much as you are able," and His saying: "And He has not placed upon you in the religion any difficulty."