Substantial amount of green leaf biomass accumulates during the post-monsoon season, which could be used for some productive purposes like fermentation, fodder and green manure etc.
Your question on "Why banyan trees (Ficus benghalensis) shed green leaves during Sept.-Nov. in southwest India?" could be answered from biological and evolutionary points of view. Plants shed their leaves in autumn in temperate lands in preparation for winter, and in the tropics, to conserve moisture which they lose through the stomata in the leaves during the dry season.
This tree in southern India which is in the tropics likely originated in the seasonal area of the warm - temperate or temperate lands. In the equatorial rain forests where the dry season does not last long, a number of such trees, like the Iroko exhibit this deciduous nature, still shedding their leaves, and surprisingly not during the dry or harmattan seasons, but towards the end of the rainy seasons, between September and October may be up to November, and with the new foliage comes the fruiting.
They seem to be displaying their evolutionary deciduous origin in their new place of habitation where there is no need for it.They crossed into the tropics during a certain period of geological history, commonly during the time of uplifts of continents and climate change mostly the Oligocene-Pleistocene.
Also check this paper: Ampaiwan, T., and Churasiri, P., Kunwasi, C., 2003. Palynology of coal-bearing units in the Mae Ramat basin, Tak Province, northern Thailand: implications for the paleoclimate and the paleoenvironment. The Natural History Journal of Chulalongkorn University, vol.3, No.2, pp.19-40.