The Mongols generally abstained from imposing religions. In addition, while in OTL, Tibetan Buddhism was introduced to China and did receive the most imperial support of all the Buddhist sects, Buddhism as a whole was accepted by the Chinese anyways by the Yuan Dynasty. I see it unlikely that the promotion of one minor sect of Buddhism could provoke the reaction asked. (Also, most Mongols still followed shamanistic principles until about the 16th and 17th centuries, so they might get annoyed too.)
e. But what if we go a little bit further back into Chinese History, and come up with a more protracted anti-Buddhist persecution under Wuzong? We can have the Tibetans build their empire earlier than OTL. We can see them invade China under the pretext of protecting Buddhists. Tibetans can do better than they did OTL, and the War in China can take a form of a Tang vs. Tibet Civil war, with the two factions espousing Taoism and Buddhism respectively. A third, radically-rationalist group can Arise out of the Chaos, and things can progress from there.