I don't know how helpful it is to your actual question, but I think The secret history of the Mongol Queens by Jack Weatherford might give you more information on the Mongol side of things. It explores the limits of trade, supply and administration in some detail and might help you to balance Mongol capabilities against what you can learn about Christendom from other sources.

Follow up question: how accurate is the book in general? I know this subreddit has a low opinion of pop history authors in general, does this book fall into the same trap?