We address the problem of calculating finite-temperature response functions of an experimentally relevant low-dimensional strongly-correlated system: the integrable 1D Bose gas with repulsive δ-function interaction (Lieb-Liniger model). Focusing on the dynamical density-density function, we present a Bethe Ansatz-based method allowing for its accurate evaluation in finite but large systems, over broad ranges of momenta, frequencies, temperatures and interaction parameters which are difficult to access using other methods. This allows us to quantify the reshaping of the zero temperature critical behavior by thermal fluctuations, in experimentally accessible regimes.