There are records of many inter-faith debates (disputations) between Jews and Christians even up to modern times. However I have never heard of similar such debates between Jews and Muslims. Are there records of such debates ever having taken place? Are there transcripts?
According to the historian Bernard Lewis, The Jews of Islam :
In general, Muslim polemicists pay little attention to the relatively insignificant Jew. Insofar as they deign to discuss the superseded religions, they are far more concerned with the Christians who, as the bearers of a competing proselytizing religion and the masters of a rival universal empire, offered a serious alternative and therefore a potential threat to the Muslim dispensation... Despite the condemnation of Jews and Judaism in the Qur'an, and in both commentary and hadith, anti-Jewish polemic was rare.
This interesting research paper in the Encyclopedia of Jews in the Muslim world discusses the historical account of the debates and polmeics between muslims and Jews. I could find some debates which were based on Alī ibn Mūsā al-Riḍā(d. 818) a prominent Muslim scholar, reportedly engaged in a public disputation with a Christian patriarch and a Jewish exilarch that is recorded by Abū Jaʿfar Ibn Bābawayh (d. 991/92) in his Kitāb al-Tawḥīd and his ʿUyūn Akhbār al-Riḍā. The arguments and modern citations to the verses mentioned in the debate are discussed here.
Most of the debates and polemics were written in the form of books, with most of them referring to both Jews and Christians, Famous scholars of important works are Ibn Ḥazm (Andalusia, d. 1064), al-Qarāfī (Egypt, d. 1285), and Ibn Qayyim al-Jawziyya (Syria, d. 1350) mentioned because of the wealth of polemical detail they provide and because of the influence they had on the development of the genre. The most important tracts by these authors are Kitābal-Fiṣal fī ʾl-Milal wa-l-Ahwāʾ wa ʾl-Niḥal, Kitāb al-Ajwiba al-fākhira ʿan al-asʾila al-fājira, and Hidāyat al-Ḥayārā fīAjwibat al-Yahūd wa-l-Naṣārā respectively, The polemics center around the theme of Peoples of the Book like in the Quran, refering both the Ahle Kitab as liable for distortion in meaning and understanding among the masses, hiding testimonies, refusing to accept the abrogation of their scriptures and laws, displaying excessive reverence toward religious Rabbis and so on.
Also, most of These written polemics have in fact been evolved from oral disputations between Jewish Rabbis and Muslim scholars.The Muslim scholar Ibn Saḥnūn of Qayrawan (d. 869) is said to have held a disputation with a learned Jew in Egypt that lasted several hours. It was initially private in nature, but turned into a public event as more and more locals flocked in to witness the account.
More ever during the Islamic golden age, Majlis sessions in which Jews were invited, or at times even summoned, to participate and to argue for their religion were held either in the salons of leading intellectuals or under the auspices and in the presence of members of the ruling elite, such as viziers, governors, sultans, and even caliphs. The famous Fāṭimid vizier Ibn Killis (d. 991), who was himself a convert from Judaism took much interest in arranging such sessions in which mutakallimūn from different faiths participated. The works of Jewish theologians like Samuel ben Hophni and Yūsuf al-Baṣīr (who are both known to have participated in such sessions) and other Rabbanite and Karaite theologians were in fact written as manuals or guides for participants in such sessions.
The Quran itself is a reflects the various debates between Jews and Christians and many of the verses reflect on the debates between the prophet and Jewish rabbis.In most of the debates the arguments are concerned with the very foundation of the Jewish faith, namely the Torah. And the Jewish Rabbi most often does not bring about rabbinical sources in the arguments.
Here is a video which enacts the debate documented in the above source, but it should be taken with a grain of salt as it is too localized and is targetted to a persian Shiite audience:
According to this article, at Jewish Virtual Library, there were instances of polemical attacks between Jews and Muslims in a literary format, but no public disputations. The two main Islamic practitioners of such polemic are said there to have been Ibn Ḥazm and Samuel al-Maghribī, the second of whom had converted from Judaism. While the article does mention the existence of Jewish response, it doesn't record the names of their authors.
Evidence, however, that there may have actually been at least one instance of a public debate between Jews and Muslims can be found in an early 13th century Persian text, housed in the British Museum. If you have a log-in for JSTOR, you can read about it in this article by Moshe Perlmann. It speaks of a public disputation between Sayyid Muhammad Mahdi Tabataba'i and the Jews of Du-l-Kifl, in the year 1211. It is an Islamic text, and it presents the Islamic arguments stronger than the Jewish ones. The disputation closed when the Muslim scholar likened the manner in which Moses superceded Abraham for the Jews to the manner in which Mohammad ultimately supercedes Moses. He then asks the Jew whom he was questioning whether he has "given thought to the pursuit of truth and knowledge", at which point the Jew requests time to think some more about his answers and the debate is brought to a close.
Shimon's answer alludes to a debate which got me interested to research more about the transcripts of the actual debate and after some research I found it in an academic paper as it appears in the translated form , the citation for this Journal paper is :
A Shii-Jewish "Debate" (Munazara) in the Eighteenth Century By: Moreen, Vera B. | The Journal of the American Oriental Society, October-December 1999
As the OP asks for transcripts of such a debate, there is a verbatim transcript from the paper of the debate between Sayyid Muhammad Mahdi Tabataba'i and the Jews of Du-l-Kifl here.