Is it permissible to vocalize during the post-Isha Sunnah prayer when a worshiper joins with the intention of performing the obligatory Isha prayer from the right, or must one keep silent?

The most authoritative opinion, supported by evidence, is that it is permissible for someone performing a fard (obligatory prayer) to be led by someone performing a nafl (supererogatory prayer). Therefore, you should do everything the Imam does, including reciting aloud and other actions. The Hanafis, Malikis, and one narration from the Hanbalis, are of the view that it is not valid for someone performing a fard to be led by someone performing a nafl. However, the most authoritative opinion is the former, based on what is narrated in Al-Bukhari and Muslim that Mu'adh used to pray the Isha prayer as a fard with the Prophet (peace be upon him), then he would go and lead his people in their fard prayer. So, it was an obligatory prayer for them, and a supererogatory prayer for him.