Overfitted neural video codecs offer a decoding complexity orders of magnitude smaller than their autoencoder counterparts. Yet, this low complexity comes at the cost of limited compression efficiency, in part due to their difficulty capturing accurate motion information. This paper proposes to guide motion information learning with an optical flow estimator. A joint rate-distortion optimization is also introduced to improve rate distribution across the different frames. These contributions maintain a low decoding complexity of 1300 multiplications per pixel while offering compression performance close to the conventional codec HEVC and outperforming other overfitted codecs. This work is made open-source at https://orange-opensource.github.io/Cool-Chic/