Is it permissible to continue communication with my paternal cousin's daughter, whom I became a godparent to before my conversion to Islam, knowing that I cannot explain my situation to her or her family?

A Muslim's interaction with non-Muslims is a virtuous deed for which he is rewarded, especially if he intends to display the merits of Islamic ethics, as Ibn Taymiyyah did in his letter to Sarjuwan. A Muslim loves good for all people and seeks reward in his benevolence towards them. The term "godparent" (al-arrab) was historically a Christian religious term describing one who undertakes the Christian upbringing of a child, but it has now lost its religious dimensions and has come to mean "guardian" or "mentor." There is no objection to continuing communication under this description, based on the principle of showing kindness to creation, while necessarily informing the girl and her family that you are Muslim. If the relationship takes on a Christian religious dimension, these aspects should be avoided, and communication should be limited to general ethical interaction.