Recent studies demonstrate the effectiveness of Self Supervised Learning (SSL) speech representations for Speech Inversion (SI). However, applying SI in real-world scenarios remains challenging due to the pervasive presence of background noise. We propose a unified framework that integrates Speech Enhancement (SE) and SI models through shared SSL-based speech representations. In this framework, the SSL model is trained not only to support the SE module in suppressing noise but also to produce representations that are more informative for the SI task, allowing both modules to benefit from joint training. At a Signal-to-Noise Ratio of -5 db, our method for the SI task achieves relative improvements over the baseline of 80.95% under babble noise and 38.98% under non-babble noise, as measured by the average Pearson product-moment correlation across all estimated parameters.