Where is Zakat al-Fitr to be paid, and is it permissible to pay it twice, when fasting three weeks of Ramadan in a foreign country and the last week in the home country, knowing that the home country has more poor people?

Zakat al-Fitr follows the individual, so the accountable person pays it in the country where it became obligatory upon them and where they are located. Thus, if the time for its obligation arrives while you are in your home country, pay the zakat there. If the time for its obligation overtakes you while you are not in your home country, then pay it there. It is not permissible to pay it except one or two days before Eid. It is permissible to transfer it from one country to another for an overriding benefit.