By the time the Mongols were rising, the church had split into Eastern Orthodox and Western Catholicism. If you want Catholic Mongols, you'll need to avoid that. Of course, if you don't care what kind of Christian they are, your best bet is probably Nestorian or Manichaean (not sure if this counts as Christianity). Perhaps the Mongols convert to Christianity for the same reason the Khazar nobility converted to Judaism? I just don't see a good reason, as the Christians were largely unable to stop the Mongol advance, and Catholic Europe was full of ignorance, poverty, and bad sanitation, if any.
"What do the Mongols gain by converting?" is the question you need to ask.
This is probably an uneducated guess, yet with such a large empire could they claim the title of Caesar for the Great Khan? Or at least it's Mongol equivalent. Or maybe they could claim to be the "Khanate-i-Romyan" or the Khanatye of Rome
Why would they want that title? Pretty much no-one outside Europe would care about it.
By the time the Mongols were rising, the church had split into Eastern Orthodox and Western Catholicism. If you want Catholic Mongols, you'll need to avoid that. Of course, if you don't care what kind of Christian they are, your best bet is probably Nestorian or Manichaean (not sure if this counts as Christianity). Perhaps the Mongols convert to Christianity for the same reason the Khazar nobility converted to Judaism? I just don't see a good reason, as the Christians were largely unable to stop the Mongol advance, and Catholic Europe was full of ignorance, poverty, and bad sanitation, if any.
"What do the Mongols gain by converting?" is the question you need to ask.