Unlike single images, capturing bursts enables many possible downstream tasks (e.g. superresolution, HDR enhancement) due to the rich information preserved in the consecutive frames. Efficient compression of these bursts is therefore essential given the additional frames to store. In this paper, we propose a novel near-lossless compression method that can preserve the most relevant information in the burst to enable multiple downstream image enhancement tasks, while at the same time reducing the file size. Specifically, we propose a two-bitstream near-lossless compression pipeline that controls the image-space distortion at frame level, and introduce the Lipschitz condition to bound the task-space distortion at burst level. Experiments conducted on a real-world burst dataset confirm the benefit of the proposed solution in terms of rate-distortion both in the burst frame space and the superresolution task space, a popular downstream task in burst processing.