The Kalmyks already fulfill your challenge, and didn't even need to convert to Christianity. But since they live on the eastern edge of Europe, that might be an issue. I think if more Kalmyks had fled China from the impending fall of Dzungharia (or perhaps if a group of Kalmyks didn't migrate back to China), the modern territory of the Kalmyks could be bigger. There's also the potential they could've arrived in one of the best parts of Europe for a steppe horde, Southern Ukraine, or "the Wild Fields" as it was called back then, and set up a state there, playing off Poland-Lithuania and Russia until it was too late. They might've had to convert to Christianity at that point, or could've stayed Buddhist despite being conquered, and later re-emerged in the age of nationalism as a Mongol-speaking (well, related language to Mongol) state in Europe. The problem with getting them there is they'd have to punch through an area well-populated by Cossacks and others, but with luck they might've been able to.