Is zakat due on merchandise that has not been in one's possession for a full week by the time zakat is due, given that the due date for zakat is the first of Ramadan each year?

What you are doing, which is paying zakat on all merchandise at the end of the year (hawl), even if you bought it one day before the year ends, is correct. This is because the profit from trade follows its origin in zakat, a point on which scholars have agreed. If the merchandise was purchased with funds obtained from sources other than trade profit, then a separate hawl is calculated for it, beginning from the time the funds were acquired. It is permissible to pay zakat on all of it at one time, at the hawl of the original funds. Whoever owns a commercial shop must pay zakat on all his money designated for trade, and any other zakatable money he possesses, at the end of the year, as long as it does not fall below the nisab. The statement attributed to Sheikh Al-Albani, that the value of merchandise is calculated randomly, is incorrect. Rather, its value must be assessed at the market price at the time of assessment, and then its value is paid as zakat.