Hi there! I don't know much about the Qianlong court itself and can't answer your question directly, but I can try to analyze the information you've given me and compare it with my knowledge of Tibetan history and hope it's useful.

Yeolha Ilgi, a diary written by a Korean ambassador to Beijing in 1780, records two discussions he had concerning "the Living Buddha." (explicitly said to be a lama from Tibet; maybe the Dalai Lama though I wouldn't know)

First, there's an interesting article concerning Tibetan-Korean historical relations. Which is very geared towards the idea that Koreans simply thought that Tibet was just the western barbarian fringe of China. (Your speciality being Late Joseon, you'll probably read more information in this article than I could.) So it's important to note that the Korean ambassador to Beijing would probably have a better knowledge of what Tibet is and who Tibetans are in 1780 as opposed to your average Hanyang-ite, but my guess is probably not by much.

The ambassador's use of the Chinese term "Living Buddha" (活佛) is (at least for me) further evidence of the "viewing the world through a Chinese lens" that seems rather common, especially where places west of China are concerned. This is an important focal point because the Tibetans themselves can be very defensive about this linguistic difference. Tsering Shakya, a Tibetan historian, in The Struggle for Tibet, his literary debate with Chinese scholar Wang Lixiong, points out that "Living Buddha" is not a concept recognized in Tibetan Buddhism and is a purely Chinese invention used to understand the Tibetan tulku institution. (It seems rather... petty, or it did when I first read it, but Shakya's argument is twofold and reasonable in retrospect: 1. For the Chinese to say "Living Buddha" versus "tulku" is not just a failure of translation, but also theologically ludicrous. There aren't "Living Buddhas" because it would also imply that there are "Dead Buddhas" when in actuality, by their very nature Buddhas don't "die" like the rest of us do, but enter "parinirvana." The same thing to sceptics and nonbelievers, like the atheist Chinese Communist Party, but a crucial distinction for Tibetan Buddhists. Especially since no tulku makes the claim to be a Buddha. And 2. In classical colonial reasoning, just because you name a concept, does not mean at ALL that you understand it, which is very true here.)

Anyway, the Lama here is most likely not the Dalai Lama. For the Dalai Lama to leave Tibet was often regarded as an evil omen, to be undertaken under only the most dire of circumstances. The only time I can think that the Dalai Lama left Tibet unprompted would have been Lobsang Gyatso, the Fifth Dalai Lama, who travelled to Beijing at Emperor Shunzhi's invitation.

The Eighth Dalai Lama would have been in his early 20's around 1780. That said, if there are Lamas in the court of the capital, they are probably from the Geluk school of Tibetan Buddhism as that would be the school patronized by the Qing Dynasty. It doesn't exactly nail down the identity of the "Living Buddha" the Korean ambassador meets.

In the first, the author talks with a Chinese scholar who emphasizes the magical power of the Living Buddha, such as his magic mirror that showed the nature of people it reflected on. The scholar also claims simply looking at the lama could cure people of evil.

Just quickly on the difference between a "Lama" and a "tulku."

"Lama" just means teacher. And all that is really required of a Lama is that they have followers. But typically they are monks.

A "Tulku" on the other hand is a unique individual. What makes a tulku different is that they (or someone else) claim they are the reincarnate of a previous Lama or saint. For example, the First Dalai Lama was claimed to be a reincarnate of the famous Lama Drom. And each subsequent Lama is the reincarnate of the previous. Tulkus are typically monks as well, but there is nothing stopping them from choosing not to take up the monastic life, as exemplified famously in the life of the Sixth Dalai Lama.

So it's important to note that Lamas are not tulkus, tulkus are not Lamas, yet quite often there are tulkus called "Lamas," ex. Dalai Lama, Panchen Lama, etc.

And while both can have supernatural abilities, it is probably more common for a tulku to make this claim than a Lama. And such claims are not made wantonly in the court of Emperors or Kings which makes me wonder even more who the Lama could be. To claim to "cure people of evil" is a pretty lofty claim, even in the Himalayas, which only deepens the mystery of the man's identity for me.

In the second, another Chinese scholar claims that the Living Buddha is a gluttonous charlatan (writing the character 貪 covetous to emphasize his point), a fat man that eats everything, including pork, beef, camel, dog, even entire donkeys, although, it is conceded, he is not interested in women.

The tulku system is pretty controversial for these sorts of reasons. It's very open to abuse (yet so are most all systems of power). It's just as common to find monks who are the opposite of this one: very interested in women, but spartan in their diet and respectful of the Vinaya code.

I'm going to make a bit of an assumption that the ambassador is... exaggerating a bit when he says the man can eat entire donkeys. But it probably also points to the meat-filled diet of Tibetans (living on the harsh Tibetan Plateau) as opposed to the more vegetarian-inclined diet I've had at Korean Buddhist temples.

However, the author finally notes that he did not encounter another man in China so opposed to Buddhism.

I'm not sure how to take this. Does the author mean that the Lama he encountered was anti-Buddhist, as in pro-tearing down Buddhist temples in China"? Or was he "anti-Buddhist" in the poetic sense, in the manner that (and bad example) a bad Priest's behavior would turn others away from the Church?

So, which view would have been more prevalent within Beijing at the time?

It's hard to say. The Manchu had some measure of control over Tibet, steadily increasing from when the Fifth Dalai Lama accepted a Qing invitation to the capital in 1652. The Tibetans mark this as the beginning of the Manchu patronization of the Geluk school. In 1720, Tibet was ruled by the Dzungar tribe who captured Tibet forcefully. The Manchu were invited at Tibetan behest to oust the Dzungars which they did and then maintained a sort of suzerainty-relationship with Tibet (this is a pretty controversial issue due to current events). But it's at least obvious by now that the Manchu at least see Tibet as their sphere of influence.

However, they weren't quick to treat Tibet as their own territory. In 1788 the Gurkhas of Nepal would attack Tibet with the intention of plundering Lhasa. The war dragged on for two years while the Tibetans beat back the Nepalis as effectively as they could, but the Gurkhas were getting pretty close to their goals. The Qing sat comfortably away and probably weren't even considering this conflict of any real concern.

In 1790 peace talks between the Tibetans and the Gurkhas broke down again and war resumed. This time, the war weary of Tibetans were worried that the Nepalis might actually reach Lhasa and requested Manchu assistance. The Manchu sent an army to Tibet and soundly defeated the Nepalis in a series of campaigns. This time, when they left Tibet in 1792, they left behind two ambans (representatives) to govern Tibet in line with Manchu rule, while leaving the country with a high degree of local autonomy.

These events were still a decade away so it's hard to say what the general attitude in China, Beijing, or just the imperial court, might be towards Tibet, Tibetan Buddhism, or what might have been known about the "Living Buddhas" among the Chinese at large.

My educated guess is that most Chinese (and by extension Koreans) if they encountered anything Tibetan would regard it as barbarian by nature, but being confronted with say, a Buddhist statue, could also acknowledge the religiosity of the people and at least find some common ground even if the language was unrecognizable. I don't know what the political consciousness was like in China in general, so I have no idea if your average, middle, or upper class Chinese person in 1780 would think of Tibet as Chinese or foreign, most likely as a barbarian tribe similar to the Mongols, having strange culture, language, and religion, if they had any thoughts at all.

The tulku institution is completely alien to Buddhists outside of Tibet (and India to a much lesser degree) so it's not surprising to find misinformation about it. Nor - and you'll probably be a little more enlightening on this than I would be - does it seem farfetched to me that a Joseon official would exaggerate the perceived evils of Buddhist priests given Joseon's traditional anti-Buddhist behavior.

But that's my 2 cents on the piece, at least. Is the book in English? I'd be very interested in taking a look myself if you could find it.