Is modifying one's sitting posture while the Imam is supplicating during the Friday sermon—specifically, sitting cross-legged with the head bowed down, looking at the place of prostration—considered an innovation (bid‘ah) or the introduction of something not part of the religion?

There is no specific posture for sitting while listening to the Friday sermon. No particular sitting posture has been prohibited except for al-ihtiba' (sitting with knees drawn up and arms clasped around them, or a garment wrapped around the body and knees). Modifying one's sitting posture when the Imam supplicates during the Friday sermon (by sitting cross-legged, bowing, and looking at the place of prostration) is not considered legally forbidden or a prohibited innovation, as long as it is not believed to be a Sunnah (definition) that must be followed. It is preferable to face the preacher and maintain eye contact throughout the sermon, whether during the supplication or otherwise.