I saw your question last night, and was like "Oh! I wrote a paper about that for a class recently!" TL;DR: The military orders were just too much of a threat. Not only were they very proficient soldiers, but they were also fanatically devoted to their cause and refused to negotiate, co-exist, or be bribed. As a pious Muslim engaged in Jihad, Saladin could not afford to spare such fierce enemies.

While Saladin was a fierce battlefield commander, he was generally merciful towards captives. However, after the Battle of Hattin in 1187, all representatives of the monastic military orders (the Templars and Hospitallers) were executed after their capture. Several factors contributed.

First, you have to understand that Saladin was a rigorously pious Muslim and absolutely committed to the Jihad against the Crusader states. Winter cited several key traits as evidence of Saladin’s strong Muslim faith. First, his adherence to Islamic law and personal piety; his faith is described as “Free of the taint of heterodoxy. Shaykh Qutb al-Din al-Nisaburi compiled a catechism (‘aqida) for his use, and Saladin studied it and taught it to his children…Saladin believed in the resurrection of the dead and hated philosophers (falaifa), heretics, and materialists. He enjoyed listening to the recitation of the Qur’an and was often moved to tears by it.” Second, Saladin loved to study the hadiths, or sayings of Muhammad; no doubt his appreciation grew from his love of Qur’anic scripture. Winters claimed that “Saladin went to such unprecedented extremes in his admiration of hadith that he ordered hadith texts to be recited on the battlefield,” and that “He sought the company of men who were renowned authorities on hadith in order to study it directly with them.” Additionally, Saladin was admired for his patronage of Musim Sufi mystics, although Winters pointed out that “Although Saladin was favorably inclined towards the Sufis, he was only a sympathetic outsider and never actually participated in their rituals."

Finally, Saladin was a champion of Sunni Islam, which was closely tied to his fervor for jihad. The two were linked because for Saladin, it was necessary to drive out the Christian invaders in order for true, Sunni Islam to be followed properly. F. E. Peters argued that therefore

The Counter-Crusade was really two movements forged into one: a Sunni counteroffensive against the rising tide of Shi’ism in the Islamic world, and particularly the aggressive, anticaliphal brand being propagated by the Fatimid dynasty in Egypt, yoked to the revival of an older notion of a jihad, a holy war waged against the infidels who presently held not only Jerusalem but the entire Muslim Holy Land of Palestine.[4]

Winter noted that more than his other religious beliefs, “Saladin’s zeal for holy war (jihad) is the aspect of his religious image that has been most subject to scrutiny." For Saladin, not only did the Prophet command him to fight for the preservation of Islam, but jihad was part of his personal identity. Robert L. Tignor argued that “Saladin saw himself as a man engaged in holy war or jihad in defense of Islam’s heartland.” Rather, “Saladin saw the decline of Islam manifested in its loss of Jerusalem…to the Christian Crusaders and set about to revive the Islamic spirit and to establish a single united empire, not under his own rule, but under that of the Abbasid caliph.” Winters summed up Saladin’s service to Sunnism as one who united various countries and groups of people under the ‘Abbasid’s yellow colors, and wrote: “Saladin not only succeeded in bringing Egypt back to the Sunni-‘Abbasid fold after two centuries of Fatimid rule; he also created a superstructure of Islamic institutions and an atmosphere of vigorous Sunni Islam, sure of itself and therefore reasonably tolerant of the religious minorities.”[8] One must assume that Winter spoke of minority Muslim sects.

At the Battle of Hattin, Saladin’s army captured most of the leaders of the Christian army, including King Guy, Gerard of Ridefort, and Reynald of Chátillon, unharmed; Count Raymond broke through Muslim lines during a cavalry charge and escaped. Foot soldiers were sold into slavery, and most knights, including Guy de Lusignan and Gerard of Ridefort, were imprisoned and eventually ransomed.

Reynald of Châtillon and the knights of the Templar and Hospitaller military orders, however, suffered a worse fate. Reynald was executed by Saladin in accordance with the former’s oath following Reynald’s attack on Muslim caravans. Ibrahim writes that “He was only the first; every other dedicated foe of Islam was denied ransom and consigned to slaughter.” Barber estimated that around 230 Templar and Hospitaller knights were summarily beheaded by Saladin’s troops as Saladin watched joyfully. William Hamblin conceded that Saladin’s apparent bloodthirst “stands in stark contrast to his generous treatment of prisoners captured later that year at Jerusalem, where Saladin paid the ransom for thousands of poor Christians and let them go free.” However, he also noted that “such atrocities by both sides were hardly uncommon during the Crusades; Richard Lionheart’s massacre of 2,700 Arab prisoners—plus their wives and children—following his capture of Acre comes to mind.” He also suggested that Saladin’s treatment of the captured Templars and Hospitallers was due to both their military competence, as well as their absolute spiritual commitment to the Crusade. He wrote

[T]he Military Orders were a threat not only because of their military strength but because of their absolute spiritual devotion to their cause as well. And that devotion, when it entailed the destruction of Islam, represented an unacceptable threat to Muslims in the age of Saladin…Finally, added to the military threat and intransigence of the Orders were the spiritual threat to Islam they represented and the blasphemous impurity they were seen to have brought to the Muslim holy places they controlled.

The footnotes and citations didn't really transfer over; sorry about that. Hopefully, this answers your question! The materials used are listed below.

Hamblin, William J. “Muslim Perspectives on the Military Orders During the Crusades.” Brigham Young University Studies 40, no. 4 (2001): 97-118.

Ibrahim, Raymond. Sword and Scimitar: Fourteen Centuries of War Between Islam and the West. New York: De Capo Press, 2018.

Winter, Michael. “Saladin’s Religious Personality, Policy, and Image,” in Perspectives on Maimonides: Philosophical and Historical Studies. Kraemer, Joel L., ed. Liverpool: Liverpool University Press, 2008.

Tignor, Robert L. Egypt: A Short History. Princeton: Princeton University Press, 2010.

Peters, F. E. Jerusalem: The Holy City in the Eyes of Chroniclers, Visitors, Pilgrims, and Prophets from the Days of Abraham to the Beginnings of Modern Times. Princeton: Princeton University Press, 1985.