I'm sorry but this idea is pretty stupid, no offense intended OP. How many Mongol battles were fought under the shade of a nearby mountain? None, the Mongols were a steppe people and lived on the plains.
Clearly, Alatau, Altai, Tian Shan, Urals, Caucasus, Hindu Kush, Tatras, Carpathians and the Himalayas are not mountains...because they conquered all of those places. Just sayin'.
Building a launching tower? Or a catapult? Ridiculous. Elementary physics says that a glider must fall a certain altitude before it gets enough of an updraft to start gliding. The Mongols weren't exactly known for their cutting-edge scientific breakthroughs, and hated any kind of technology that couldn't be carried at the speed of a galloping horse, because their military campaigns existed on their capacity for fast moving. Are they really going to invent a 200 metre tower that can be dragged at the speed of their horses?
That's...just...wow. Do you know what made Mongols Mongols, instead of being mere Cumans or Pechenegs? Their ability to take formidable cities by siege or storm in weeks at most. And they probably brought the trebuchet to Europe. Even more convincingly for this argument, it got elaborated on really quickly in the West, and then taken in its ultimate counterweight form back to China to siege down the fortresses in Sichuan by the very same Mongols that hated technology and sieges - disseminating the ultimate medieval siege machinery west and then back east across all of Eurasia in less than 100 years.
If they were just dedicated horsemen and nothing more, they wouldn't have built the vast Empire they did, because good walls stopped many like them before repeatedly. Except this one time.
Finally, nobody travels at gallop, and Mongol horses weren't really strong or fast. What they were was hardy and numerous. Mongol strategic speed was due to organisation rather than some kind of miraculous four-legged superweapon.