Building ASR systems robust to foreign-accented speech is an important challenge in today's globalized world. A prior study explored the way to enhance the performance of phonetic token-based ASR on accented speech by reproducing the phenomenon known as interlanguage speech intelligibility benefit (ISIB), where foreign-accented speech is more intelligible to listeners sharing the speaker's native language than to native listeners. ISIB was technically implemented by using the speaker's L1 to learn k-means cluster centroids in an SSL feature space to obtain phonetic tokens. In this study, we propose a more advanced modeling of ISIB. By employing differentiable k-means and optimizing the entire module for both L1 and L2 ASR, the proposed method outperformed the baselines, both when using only native speech and when additionally incorporating a limited amount of accented speech. Notably, in the latter scenario, our method achieved approximately a 20% relative improvement in recognition accuracy.