would take inspiration from Toyotomi Hideyoshi. The besieging force basically made a town of there own outside Odawara complete merrymaking just to let the Hojo hear it, they surrendered after 3 months. Since in Japan there is an absence of siege weapons and siege tactics are at best wait until who gets to run out of supplies first or weather out the expected sally out attack regardless of odds.
Like 10 years (Ishiyama Honganji), reduce the besiged to cannibalism, find a way to counter the shrewd besieged commander's idea of planting fruit bearing trees near the walls and dried vegetables waved in tatami mats. So in game terms, it translates to at least 10 turns in the campaign map of waiting for some damn siege to finish and hope to Hachiman the commander is itching for a sally out and then you can get the killing blow. Not really a nice way in gameplay.
No seriously, the siege weapon you call mangonels that hurl thunderclap bombs only appear at the Onin War, thats like the 15th century, so their CA made the preogative to made them available coz yeah to the uninitiated it would not make sense. During the initial invasion during the Imjin War, they would be facing different enemies, different modes in warfare, different castles, yet the Japanese never had a siege train to accompany it.*
*Osprey Siege Weapons of the Far East 2.
I know Wiki might not be a good source but whatever.
There is no record that the goal of destroying walls ever entered into the strategy of a Japanese siege. In fact, it was often seen to be more honorable, and more tactically advantageous on the part of the defender for him to lead his forces into battle outside the castle. When battles were not resolved in this way, out in the open, sieges were almost always undertaken purely by denying supplies to the castle, an effort which could last years, but involved little more than surrounding the castle with a force of sufficient size until a surrender could be elicited.