To control bloodshed, you must decrease the desirability of the crown to those who are in line to receive it.
You can do this in two ways.
- Make the King's life shorter and less enjoyable.
- Make the lives of those waiting to be King more enjoyable and filled with privledges.
Add a rule that no King may wear the crown for more than ten years and that every King must die painfully on the tenth anniversary of their inauguration.
Add another rule that the King must lead every army into battle from the front line, and must therefore train for long hours every day as a warrior.
Grant unlimited privledge to ALL of the heirs, giving them the best clothes and food, their choice of the best concubines and freeing them from any battle duties or training. Let them live in lavish luxury and in silent terror of the day when they must become king.
Set all these rules in stone so that no future king may change any of them.
You will have the King's heirs all carrying first aid kits rather than swords, when they visit their older siblings.