First let me preface this by stating that the definition of rationalism you're working with is very much framed by European early-modern sensibilities: i.e. the way you're framing rationalism as the rejection of religious thought. Whereas if you look at earlier periods, again in Europe, you'll find that notions of rationalism are actually quite intertwined and commonly used in even Christian theological senses (like with Thomas Aquinas). That is to say, I'd just like to clarify that there's not necessarily the dichotomy of religion vs rationalism that you may or may not be assuming, historically.

As for what I can speak to more specifically, the most prominent example in China that comes to mind is the school of Moism/Mohism which essentially dropped out of existence as a separate entity by the end of the Warring States period. One of the major pre-Qin philosophical schools, Moism existed alongside Confucianism (which emphasized social/familial order), Daoism (which emphasized, as a poor summary, cosmic/mystical order), and Legalism (which emphasized the order of the state and its ruler). The Qin dynasty famously emphasized the Legalist school of thought during its short existence, suppressing the the Confucian and Daoist schools for that period, while the Moist school itself largely just sort of puttered out during the period. However, it's important to note that just because Legalism and Moism no longer could really be counted as separate branches after the Qin and Warring States periods accordingly, that doesn't mean that their philosophical ideas totally disappeared out of existence. Tenets of both schools of thought found their way into mostly Confucian thought in subsequent years (and it's also fairly false to claim any dichotomy between Confucian and Daoism thereafter either, as individuals would most often not view them as conflicting philosophies, instead thinking of them as answering different questions/pertaining to different situations. There were however major differences between Daoist and Buddhist schools of thought later, but Buddhism isn't native to China of course).

Moism, in a very brief nutshell, perhaps can best be viewed as a contrast to Confucianism rather than Daoism. Confucianism focused on familial affiliations, and how relationships and their meanings and importance differ from family to community and onward. Close kin deserve more attention and respect than non affiliated strangers in Confucian thought. Moism however stressed that relationships at all strata existed on equal status, and no preference should be given based purely on kinship. Later on, Confucian thought actually also absorbs some of the ideas present in Moism regarding the state being structured as a meritocracy, rather than heavily based on blood ties like in the pre-Qin period. Moism also stressed that arts and music are essentially useless to governance and the populace, again in contrast to Confucian thought. Like the ideas of meritocracy, you can see here the Spring and Autumn/Warring States period roots and influence present in Moism. These were periods of heavy multi-state wars, and philosophies that reject the often-viewed-as-incompetent authorities and indulgences that were so prevalent during these harsh periods would've been extremely attractive. As those ideas got absorbed mainly into Confucianism (as some Legalist ideas also were), in one form or another, Moism lost a lot of its bite and relevance.