Is it permissible to count the one thousand dollars that I lent to a needy person who is currently unable to repay it as zakat, knowing that a full year has passed over the money and zakat has become due?

This brother's claim is not considered part of your zakat (definition) on wealth. It is as if you have waived a debt that you have not yet collected for a zakat year from your wealth, while keeping the zakat that you are obligated to pay. This money is practically non-existent because the debtor is in hardship. The obligation is either to forgive the debt or to wait until he is able to pay, as Allah Almighty said: "And if there is one in hardship, then grant him respite until a time of ease. But to forgo it as charity is better for you, if you only knew."