I doubt that there is anything quite as extensive as the religious corpora you mention. You'll find a lot of information in Holger Gzella's A cultural history of Aramaic (2015). Every stage (from Old Aramaic up to Late Antiquity) is described in a separate chapter which starts with an overview of the corpus. From most periods there are (often short) inscriptions and private letters, and from the time that Aramaic served as an international language there are also numerous legal and administrative documents.
In terms of literature, one can mention first (in terms of chronology) the Deir Alla inscription, a fragmentary text with linguistic similarities to Biblical Hebrew narrative. It makes reference to several deities and Gzella classifies it as religious literature, though I'm not sure one can draw a hard line in the Ancient Near East. You could also think of the Tell Dan inscription, but perhaps it is more historiographical than literary – again, a distinction that is hard to make.
The first large literary works appear during the rule of the Achaemenid Empire. Gzella writes (p. 201):
Elements of a formal prose style surface already in the earliest royal inscriptions in Aramaic from ninth- and eighth-century B.C.E. Syria. Songs, stories, and myths no doubt existed in popular lore as well and were passed on orally from one generation to the next. Traditional proverbial wisdom first crystallized into a surviving text some time before the rise of the Persian empire and was eventually associated with the wise counsellor Aḥiqar (see Section 3.4). The Achaemenid period then saw the emergence of literary works in some provinces, which later continued into the production of vast bodies of religious compositions that corroborated the ideas and cultural self-awareness of Jews, Christians, and Mandaeans. One may therefore suppose that national literary traditions slowly began to take on their shapes with the increasingly institutionalized use of Aramaic.
The most well-known is Aḥiqar, found in Egypt but probably widely circulated at the time. There are a couple of much shorter and/or fragmentary texts from the same period, such as Proto-Esther (from Qumran: 4Q550), which is identified with the biblical book of Esther (which could be called non-religious). From Egypt, Gzella also mentions Bar Puneš, which like Aḥiqar and Esther is probably a court novel but is too fragmentary to tell for sure, and Papyrus Amherst 63, which is liturgical and probably does not qualify for your question. (In Palestine, a similar development led to the book of Daniel.)
Later literary texts, such as Bardaisan's Book of the Laws of Countries, do assume a certain theological point of view but are not exclusively religious. This is also true of Talmudic texts. Here the distinction again becomes vague.
