Kenneth Clark states his opinion in Civilisation that nearly all Latin literature still extant today was saved thanks to Charlemagne, who instructed his kingdom's copyists to work on non-Christian Latin texts. I cannot find evidence for this online. Do historians believe this?
I would disagree and I think that most medieval European historians would agree with me (this was my minor in college).
Most copyists in the Middle Ages were members of Holy Orders established by the Roman Catholic church staring ca. 500 CE around the time of the fall of the Roman Empire and sustaining their task through the invention of the printing press ca. 1500 CE. It is far more remarkable that monks in Holy Orders copied so many non-Christian (and even explicitly pagan) works than it is the Charlemagne, the non-ordained ruler of a Christian kingdom, did so.
Charlemagne is ca. 800 CE and the number of copyists that he would have had any meaningful control over would have been modest relative to those in functionally autonomous monasteries. His traveling court would spend only a few days in each place in a circuit around his domain that could take multiple years. Importantly, his domain didn't penetrate into key source areas of Latin texts like (much of) Italy (home to many of the biggest cities in the Western Roman Empire) or Constantinople (the capitol of the Eastern Roman Empire), or into places with particularly productive monasteries like Ireland.
Charlemagne very likely did do his own small part towards the effort with his small share of the total universe of copyists at the time (and this is legitimately commendable), so Clark's observation isn't totally baseless. But, Clark overstates how important Charlemagne was in the larger scheme of things.
Also, the Byzantine Empire (i.e. the Eastern Roman Empire) was a functioning state for the entire time period that Charlemagne ruled and had direct continuity with Roman Latin sources.
Further, many non-Christian Latin texts were preserved by the Islamic empire which was thriving and growing at a time when European culture was in its darkest era.
I would say that Charlemagne and the Carolingian Renaissance that he ushered in at his Capitol, Aachen in Northwest Germany, played an important, but not necessarily a very significant role in translating and preserving Latin texts. Most Latin texts-(like many Greek texts), were translated and preserved by the Byzantines in Constantinople throughout the Middle Ages-(including during Charlemagne's time). And many Latin, (as well as many more Greek texts), were also translated by the Arabs during the Middle Ages in countries, such as Iraq, Egypt and especially, Spain.
Nevertheless, Charlemagne was perhaps the first European Ruler-(North of Italy) to help liberate much of Western Europe out of "The Dark Ages"-(albeit for a short time), though his resources, when compared with the Byzantines and the Arabs, were less plentiful and in turn, his cultural influence, though historically noteworthy, was limited.