Java is not in South Asia, but for Old Javanese Literature before explicit Islamic influence, this is the best place to classify this entry for now.
I try to read a bit about the history of Old Javanese literature because of the context it provides for Malay literature, and found and read this:
Hall, Kenneth R. "Traditions of Knowledge in Old Javanese Literature, c.1000-1500." Journal of Southeast Asian Studies 36.1 (2005): 1+.
Key Extant Texts mentioned:
- Ramayana: earliest surviving kakawin in Old Javanese, 9th c., Central Java
- Uttarakanda, Old Javanese preamble to the Ramayana
- Arjunawiwaha kakawin: between 1028-35, by mpu Kanwa, East Java, elaborates on the plot of Mahabharata
- Bharatayudda kakawin: 1157, expands upon Mahabharata
- Parthayajna kakawin: date not too clear in the article, it is the last work mentioned in the period between c.1000-1226, and says the text is basis of iconography of Candi Jago built in the last quarter of the 13th c. and rebuilt shortly before 1350, but says it is a 14th c. text. I assume the last point is in error, and that the text is before 1226. It is also a text wtih Buddhist influence - ignorance / hate / desire as three worldly poisons, and the 6 realms of samsara - but the article author does not explicit wrote of this as being influenced by Buddhism.
- Nagarakertagama: 1365, by mpu Prapanca (a Buddhist cleric) mentioned many times but not addressed directly in the article
- Arjunawijaya: c. 1370, by mpu Tantular, Majapahit (East Java), talks about unity of Siwa and Buddha; based on Uttarakanda, could be interpreted as prelude to the poet's later work Sutasoma
- Sutasoma kakawin: before 1389, also by mpu Tantular, Wisnu as manifestation of Buddha; tantric Buddhism; provides one of the most sophisticated theological statements in the kakawin literature
- Siwaratrikalpa: c. 1470, by mpu Tanakung; highlights Saivite ritual (tantric)
Just as a comparision, V. Braginsky in his work on traditional Malay literature, in the first 3 chapters, did mention the following kakawin works:
- Ramayana: one of the most important Old Javanese poems, first literary work per se in Javanese
- Arjunawiwaha: one of the most important Old Javanese poems, first typical Old Javanese literary work
- Bharatayudda: one of the most important Old Javanese poems, Braginsky says there is a later Majapahit version ~1400AD
- Nagarakertagama: tells about Javanaese Buddhist monasteries founded in South Sumatra, describes Majapahit's conquest of Pasai - (According to Cambridge History of SE Asia Vol 1 Chapter 5 p.278, this work is "a long poetical text composed in 1365 by a high official at the court of Majapahit.")
- Arjunawijaya: one of the most important Old Javanese poems
- Sutasoma: Braginsky mentions in passing. Probably too late to exert influence on Malay literature before ~1550AD
Besides the above, Braginsky also mentions other works: Sanskrit-related works (10th c. Sang Hyang Kamahayanikan, Kritabasa, Amaramala), other kakawin (kunjarakarna from west Java, Gatotkacasraya, Dewa Ruci, Smaradahana, 13th c. Bhomantaka), histories (Pararaton, 17th c. Bahad Tanah Java), Tantu Panggelaran (mythical encyclopedia, Majapahit era), and Middle (?) Javanese works related to Panji (Waseng Sari, Malat - these two the earliest Middle Javanese Panji-romances) late 16th c. Wangbang Wideya, Panji Anggreni), a shadow play related to Mahabharata (wayang purwa), and a poet who tried to date Java's mythological history (Ranggawarsita).