I am not an expert on this topic, thus, here I only would like to discuss with readers about what I found out in secondary literature and the seeming problems the secondary literature entails.

• The most comprehensive resource I could locate are the books and articles edited or authored by Peter Schalk, who appears to be the major expert on Buddhism in Tamil Nadu and claims to have examined all possible primary sources on Buddhism in that country. Through a cross-examination of coins, inscriptions, artefacts, texts on Buddhists and texts of Buddhists, Schalk could conclude that “none is before the 4th and none after the 14th century” (Schalk in Deeg et al., 2011, section 8). More in detail, the last Tamiḷ Buddhist document (an inscription displaying a syncretic form of Buddhism and Śaivism, see Āḷvāppiḷḷai Vēluppiḷḷai 2002, section 5.7) is dated to the 13th c.
• Buddhism, was, moreover, never supported by royal patronage in Tamil Nadu, unlike Vaiṣṇavism, Śaivism and in part also Jainism. Thus, it lacked the protection it could enjoy in other parts of South Asia and in Śrī Laṅkā (Schalk 2011, section 10). This, together with the pressure from Śaivism and Vaiṣṇavism are probably the causes of the Buddhist decline described as early as in 600 AD (in the Mattavilāsa prahasana, see Schalk, 2013, p. 30).
• The lack of importance of Buddhism in the intellectual arena of Tamil Nadu is also testified by the fact that Jainas are much more frequently attacked and criticised by Śaiva and Buddhist authors (see Schalk 2013, p. 33).
• In fact, even before the 14th c., Buddhism in Tamil Nadu had evolved into a form of Buddhism-Śaiva syncretism, so that:
  

  This “freedom’s” strenght was also its weakness: without an authoritative textual base it was soon assimilated with Caivam [=Śaivism] and finally eliminated in the 14th century” (2011, section 1).

  This open and syncretic nature of Tamil Buddhism is also evident in the fact that, as shown again by Schalk (2011), it did not possess a proper canon.

• A further interesting resource is Anne Monius’ 2001 book, which focuses on the problems entailed in the study of the Buddhist community in South India through texts which only imagine it, such as the poem Maṇimēkalai. I will not focus on her text here, since my main concern is with a later period (1000-1500).

What else can we say about what Buddhists in Tamil Nadu read or listened to, and believed?

1. The Chinese pilgrim Xuanzang (ca. 602–664) speaks of a large community and that it belonged to the Great Vehicle. However, Schalk convincingly shows that Xuanzang was speaking from hearsay and is not fully reliable (2011, section 11).
2. There is a widespread tradition (of which I could not locate the primary source and I ask for help from learned readers) saying that in 1236 a group of bhikkus from Kañci left for Śrī Laṅkā to re-establish there a Theravāda ordination line.
3. The Jaina Tamil text Nīlakēci seems to target a Abhidharma-like kind of Buddhism (see Āḷvāppiḷḷai Vēluppiḷḷai 2002, section 5.4, especially 5.4.8; for the identification of the Nīlakēci‘s polemical target with Mahāyāna, see Kandaswamy 1999, to which Shalk 2002, section 1.4.2, polemically replies).
4. The vīracōḷiyam treatise (written during the reign of Vīrarājēndracōḷa, 1063–1070, and commented upon in the 12th c.) is a Buddhist text on Grammar. I wonder whether it could be connected to the flourishing of Buddhist Grammars in Pāli countries (see Ruiz-Falqués’ studies thereon, here) or rather only to the Cāndravyākraṇa.

Point No. 2 might seem to slightly clash with the evidence of the Theravāda ordination line in Śrī Laṅkā being re-established by monks from Pagan (Burma). Nonetheless, it is not impossible that the ordination line was interrupted again and one needed again bhikkus from abroad. These bhikkus most probably did not belong to the syncretic Buddhism described above. In fact, Schalk explicitly acknowledges the presence of Pāli ācariyas (ācārya) in Tamil Nadu, although he adds that

We know that they were also endured in Nākapaṭṭiṇam during the Cōḷa period, but they were evidently secluded, because they left no traces in the documents produced by the Cōḷa establishment (Schalk, 2002, section 5.1.1).

Now, a last question: Do you know whether Buddhist texts were preserved in Jaina institutions in Tamil Nadu, as it happened in North India? This could account for the presence of Buddhist texts even when an institutionalised Buddhist community was absent…

If you are wondering why I am interested in the topic, you can read this post of mine on Veṅkaṭanātha’s (Tamil Nadu, 1269–1370) Buddhist quotes.