1. Using a (persisted) computed column would be my preference also. If the date portion is placed first in the compound field, you could arrange the partition function so that changing the processed status only results in a change of partition for rows older than 3 months. You would then just not archive partitions in the pre-3-month range which hold unprocessed rows. To minimize data movement when crossing the 3-month boundary, ensure that processed rows stay in the same partition as they age.

2. Assuming that there are relatively few unprocessed rows older than 3 months, another strategy might be to temporarily move the unprocessed rows older than 3 months to a temporary table or variable just before the partition is archived, and then re-insert them afterward.