The ban occurred during Mamluk period. The specifics of why are unclear in the wiki entry but the Jewish Virtual Library helpfully suggests that it's because they turned the location into a Mosque. Assuming that this was the correct sequence of events and that the ban went into effect around then, two names appear in the Al-Jawali Mosque's wiki entry:
Al-Jawali Mosque was built on the orders of the Mamluk Governor of Gaza and Palestine, Sanjar al-Jawli, between 1318 and 1320 during the sultanate of an-Nasir Muhammad.
Firstly, it's unclear when the ban went into effect exactly. I've found no immediate indicators of whether the ban occurred before, during, or after the Mosque was built; only indications that the ban occurred because a Mosque was built on the site.
Secondly, and complicating things a little more, Al-Jawali got ousted in 1320 (the year the Mosque was finished) over a private dispute with the governor of Syria, and there's no indication of whether he was ousted before or after the Mosque was finished. If the latter and the ban came into effect after the Mosque was finished, the name you're looking for would be his successor, Muhammad ibn Baktamur.