Despite recent advancements in global population well-being and food security, climate change threatens to undermine child nutritional health, particularly for marginalized populations in tropical low- and middle-income countries. South Asia is at particular risk for climate-driven undernutrition due to a combination of historical weather exposures, existing nutritional deficits, and a lack of sanitation access. Previous studies have established that precipitation extremes increase rates of undernutrition in this region, but the existing literature lacks adequate consideration of temperature anomalies, mediating social factors, and the developmentally-relevant timing of exposure. We combine high-resolution temperature and precipitation data with large-sample survey data on household demographics and child anthropometry, using an approach that incorporates three key developmental periods and a rigorous fixed effects design. We find that precipitation extremes in the first year of life significantly decrease children's height-for-age (HAZ) in South Asia. The detrimental effects of extreme precipitation are especially concentrated in under-resourced households, such as those lacking access to proper sanitation and education for women, while anomalous heat is particularly harmful for children in Pakistan, though it tends to benefit children in some demographic groups. These results indicate that nutritional status in South Asia is highly responsive to climate exposures, and that addressing sanitation infrastructure and other development priorities is a pathway towards reducing this vulnerability.