Building on work of Balakrishnan, Dogra, and Bianchi, we provide some improvements to the explicit quadratic Chabauty method to provably compute rational points on genus 2 bielliptic curves over Q, whose Jacobians have Mordell–Weil rank equal to 2. We complement this with a precision analysis to guarantee correct outputs. Together with the Mordell–Weil sieve, this bielliptic quadratic Chabauty method is then the main tool that we use to compute the rational points on the 411 locally solvable curves from the LMFDB which satisfy the aforementioned conditions. This is joint work with Francesca Bianchi