Exactly, a Sinicized Christianity would simply fit it into the religion with little or no problem. A bit stronger than the Divine Right of Kings, but not too strong.
The Mandate of Heaven isn't a religion. It's a social and political concept and tradition that dates back to the foundation of the Zhou dynasty (the oft envisaged "Golden" Age of Chinese society, as the Confucians viewed it). Neither also would political philosophies such as Confucianism count, given they're strict focus on the physical, and rejection of metaphysical elements in analyses (this changes with neo-Confucianism, but only then does it simply expand the bounds of the philosophy itself).
Both Daoism and Buddhism are major religions, however the former is a "home-grown" religion, so to speak and thus is highly receptive amongst the people and intellectuals because of its native origins, and the latter is a "foreign, barbaric" religion, which gained the popularity it did in China due to syncreticism, the turmoil and sponsorship of the various states during the Northern and Southern dynasties period, and the continued sponsorship of the Tang dynasty throughout its long reign. The establishment of Buddhism in China is a remarkable, and continuous growth by the various monasteries, and state sponsorship during a tumultuous era. In a sense, they fulfilled the same role as medieval European monasteries in being at the forefront of productive technologies (situating themselves atop mountains, and such).
If you want to learn more, pick up Lewis's
China Between Empires: Northern and Southern Dynasties. His books on Imperial Chinese history (from the Qin at present up until the Tang [just recently published]), are absolutely brilliant in describing the social, political, economic, cultural, and religious practices, developments, etc., during the periods in question, in a relatively readable and engaging text (it jumps around alot, mind, but so does the subject material he has to work with). I'm personally waiting for the rest of the series to come out (give it a few years).
Now in regards to Christianity in China, the PoD needs to be quite early. In all honesty, it's probably too late for Islam to leave a lasting mark in China. Christianity,
maybe, but is suffers from being a monotheistic religion (very difficult to spread, given the breadth of the Chinese mythological "pantheon", so to speak, whereas Buddhism can coexist with the many deities), the fact it is a foreign, barbaric religion (Buddhism is also foreign, but it was reviled during the time period of its spread as well by the intelligentsia for it, despite often times extremely heavy promotion [things like tax exemptions for example] of Buddhism by the state [from both the Northern and Southern dynasty period really through to the Yuan]). Plus add to the fact the holy sites, major religious scholars, etc. are extremely distant and thus contact is
very limited for the spread of Christianity to occur (you can have small communities of Christians living in China, as per OTL). In Buddhism's case, we have extensive interaction between developing Chinese monasteries with Tibetan, Indian (at the time), and other major Buddhist stronghold to help foster the spread, translation, and development of new religious doctrines. Any such contact between China and Europe will necessarily be limited.
So given these factors (barring things like Taiping being successful [unlikely]), it's probably borderline ASB for Christianity to develop deep roots in China.
I do think if the alternative option of the Revived Confucianism with the "face" more like Xun Zi's sects (like inherited bad human natural philosophy). Then upper class should be more acceptable to religions like Christianity with concept like Sins. Perhaps a lot of what "Confucius said" will also be different with what we know today. There were a lot editing/banning after this decision. Scholars chose certain texts instead of others to fit the bill, a lot original texts were lost. And scholars centuries later tend to seek answers very literally from "ancient texts", not knowing what the original version was. (But by now from archeology we know a lot of the so called "ancient texts" were changed due to these "editing", and are different from what we dug up before this period)
Oh
yes. One of the biggest issues you'll find with things like the Five Confucian Classics and the Four Books of Zhu Xi. With the Five Confucian Classics, modern historical dating of the various texts by comparing the linguistics of it to historical details come up with a range dating from the Zhou to the Han dynasties. The Classic of History, for example is divided into four major periods (purporting to be from the Zhou, the Shang, the Xia, and the mythical Lu dynasties respectively). It's fascinating for the various political philosophies it develops (i.e. the conflict between meritocracy v. aristocracy, the ultimate role of the state, etc.), but in terms of dating, only the Zhou period passages are probably historically accurate. The Shang and Xia passages can be dated to the Warring States period or later, and the Lu passages can be dated to the Han dynasty pretty definitively (the Lu passages are written as if by a Han dynasty scholar, based on the style and historical references and anachronisms, etc.). The Four Books of Zhu Xi are less egregious, mostly because everyone already knows they were compiled, edited, and commentated on by Zhu Xi in the Song dynasty anyways.