The only problem there is that AFAIK the point where the mongols were unified and motivated to carry out full fledged warfare instead of small raids is immediately followed by them conquering a large swath of China and incorporating their engineers and technology, and the kinds of fortifications they'd be dealing with there (heavy sloped wall earthworks) were both intrinsically resistant to siege weapons and vulnerable to large scale infantry assaults. So by the time they were waging war in eastern europe or the middle east they were both an experienced, professional military machine with excellent logistics and in possession of enough engineering knowledge to build siege weapons.

And of course it has to be said that they didn't pursue that advantage for further conquests because basically their whole motivation was conquering and unifying the various nomadic steppe peoples and their goals for waging war against sedentary kingdoms was either pursuing and subjugating other steppe peoples or responding to attacks/affronts. They weren't some conquering force of nature out to paint the map like a grand strategy game player, they had clear and concrete goals and conquering and holding sedentary kingdoms' land wasn't one of them.

So in TW:M2 they'd be at the point where they'd both have engineers, but also probably wouldn't bother with a castle unless they needed to. They also, obviously, could have dismounted and stormed a castle defended only by a skeleton crew without any trouble. IRL horse archers aren't glued to their mounts any more than knights were, and so horses could just as easily be used for infantry mobility as they could a weapons platform.